Unit 1: Polynomial and Rational Functions

Definition and Characteristics of Polynomials

• A polynomial is a mathematical expression made up of terms that are the sum of constants multiplied by non-negative integer powers of a variable. Each term is called a monomial, and the highest exponent of the variable determines the degree of the polynomial. For example, \( f(x) = 4x^3 - 2x^2 + x - 7 \) is a cubic polynomial because the largest exponent is 3.
• The degree of a polynomial affects both its shape and its end behavior. Higher-degree polynomials can have more turning points, with the maximum number of turning points being one less than the degree. This is why quadratic functions can have at most one turning point, cubic functions at most two, quartic functions at most three, and so on.
• The coefficients of the terms, especially the leading coefficient, influence the steepness and direction of the graph. A positive leading coefficient for even-degree polynomials causes the ends to rise, while a negative leading coefficient for even-degree polynomials causes both ends to fall. For odd-degree polynomials, the sign of the leading coefficient determines whether the graph falls to the left and rises to the right, or vice versa.
• Every polynomial function is continuous and smooth, meaning it has no breaks, jumps, or sharp corners. This is an important distinction from rational functions, which can have discontinuities due to division by zero. Continuity ensures polynomials are well-defined for all real numbers, making them predictable in modeling scenarios.
• Polynomials can be classified by both degree and number of terms. For example, a degree-2 polynomial is a quadratic, a degree-3 polynomial is a cubic, and so on. The number of terms also gives names like monomial (1 term), binomial (2 terms), or trinomial (3 terms), and these structures can help identify factoring strategies.

End Behavior of Polynomial Functions

• End behavior describes how a function behaves as \( x \to \infty \) or \( x \to -\infty \). For polynomial functions, the degree and leading coefficient determine the end behavior. This is often summarized in a chart for quick reference when sketching graphs.
• For even-degree polynomials with a positive leading coefficient, both ends of the graph rise toward \( +\infty \). For even-degree polynomials with a negative leading coefficient, both ends fall toward \( -\infty \). This symmetry in direction is a defining characteristic of even-degree polynomial graphs.
• For odd-degree polynomials with a positive leading coefficient, the left end of the graph falls to \( -\infty \) while the right end rises to \( +\infty \). For odd-degree polynomials with a negative leading coefficient, the left end rises to \( +\infty \) while the right end falls to \( -\infty \). This opposite direction behavior is a defining characteristic of odd-degree polynomials.
• The concept of end behavior connects directly to limits in calculus. For example, \(\lim_{x \to \infty} ax^n = \infty\) if \(a > 0\) and \(n\) is even, which formalizes the graphical observations students make in precalculus. Understanding this connection makes the topic easier later in calculus.
• When graphing, you can quickly determine end behavior by focusing only on the leading term \(a_n x^n\) and ignoring lower-degree terms. This works because as \(x\) becomes very large or very small, the highest power term dominates the function’s growth rate and direction.

Zeros and Multiplicity of Polynomials

• The zeros of a polynomial are the x-values where the function equals zero, also called roots or x-intercepts. They are found by setting the polynomial equal to zero and solving, often through factoring or using the Rational Root Theorem. For example, if \(f(x) = (x-2)(x+3)\), the zeros are \(x=2\) and \(x=-3\).
• Multiplicity refers to how many times a zero repeats. If a factor \((x-r)^m\) appears, then \(x=r\) is a zero with multiplicity \(m\). Multiplicity changes the shape of the graph at that zero—odd multiplicities cause the graph to cross the x-axis, while even multiplicities cause the graph to touch and turn around.
• Higher multiplicity values cause the graph to flatten out near the zero. For example, a zero with multiplicity 3 still crosses the axis but looks flatter compared to a zero with multiplicity 1. This visual behavior helps identify multiplicities when analyzing a given graph.
• The sum of all multiplicities equals the degree of the polynomial. This means if you know the degree and some zeros, you can deduce missing zeros or their multiplicities. This is a powerful check when constructing polynomials from given information.
• Zeros can be real or complex. Real zeros correspond to x-intercepts on the graph, while complex zeros (which come in conjugate pairs if coefficients are real) do not appear as intercepts but still influence the polynomial's overall shape and factored form.

Factoring Polynomials

Common Factoring Strategies

• Always start factoring by checking for a Greatest Common Factor (GCF). For example, \(6x^3 - 9x^2\) has a GCF of \(3x^2\), so factoring it out gives \(3x^2(2x - 3)\). Forgetting this step can make later factoring harder or even impossible without first simplifying.
• Quadratic trinomials (degree 2) can often be factored into two binomials if the product of the outer coefficients equals the constant term. For example, \(x^2 + 5x + 6\) factors into \((x + 2)(x + 3)\) because \(2 \cdot 3 = 6\) and \(2 + 3 = 5\). Always check your factoring by multiplying back out.
• Difference of squares applies when you have two perfect squares subtracted: \(a^2 - b^2 = (a - b)(a + b)\). For example, \(x^2 - 49\) becomes \((x - 7)(x + 7)\). Many students mistakenly try to factor a sum of squares, which cannot be factored over the reals.
• For higher-degree polynomials, use factoring by grouping. Split the polynomial into two parts, factor each part, then factor out the common binomial. For example, \(x^3 + 3x^2 + 2x + 6\) becomes \((x^2( x + 3) + 2( x + 3)) = (x + 3)(x^2 + 2)\).
• When factoring completely, continue the process until all factors are prime over the reals. For example, \(x^4 - 16\) factors as \((x^2 - 4)(x^2 + 4)\) and then \((x - 2)(x + 2)(x^2 + 4)\). This ensures no factoring opportunities are missed.

The Remainder and Factor Theorems

Using the Remainder Theorem

• The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x - k\), the remainder is simply \(f(k)\). For example, dividing \(x^3 - 4x + 1\) by \(x - 2\) gives a remainder of \(f(2) = 8 - 8 + 1 = 1\). This shortcut saves time over long division when you only need the remainder.
• This theorem works because substituting \(x = k\) into the polynomial evaluates exactly what’s “left over” after factoring out \(x - k\). If the result is zero, then \(x - k\) is a factor. This connects directly to the Factor Theorem.

Using the Factor Theorem

• The Factor Theorem states that if \(f(k) = 0\), then \(x - k\) is a factor of \(f(x)\). For example, since \(f(3) = 0\) for \(f(x) = x^3 - 3x^2 - 4x + 12\), \(x - 3\) is a factor. This gives you a starting point for further factoring.
• You can use synthetic division to divide by \(x - k\) once a factor is found, reducing the degree of the polynomial. For example, dividing \(x^3 - 3x^2 - 4x + 12\) by \(x - 3\) gives \(x^2 - 4\), which can be factored further. This method is faster than long division and is commonly tested.
• Always check multiple possible factors using the Rational Root Theorem before committing to division. Testing each possible root with the Remainder Theorem can quickly reveal all rational zeros, speeding up the process.

Graphing Polynomial Functions

Step-by-Step Graphing Process

• Step 1: Identify the degree and leading coefficient to determine end behavior. This gives you the basic direction of the graph’s ends, which is essential for an accurate sketch. For example, a cubic with a positive leading coefficient will fall left and rise right.
• Step 2: Find the y-intercept by evaluating \(f(0)\). This gives one guaranteed point on the graph. For example, if \(f(x) = 2x^3 - 3x^2 - 5x + 6\), then \(f(0) = 6\) means the graph passes through \((0, 6)\).
• Step 3: Find all real zeros and their multiplicities. This tells you where the graph crosses or touches the x-axis and how it behaves at those points. For instance, multiplicity 2 means the graph will touch and turn, while multiplicity 1 means it will cross.
• Step 4: Plot additional points between and beyond the zeros to determine the curve’s shape. This is especially important for higher-degree polynomials with multiple turns. Use symmetry if applicable to reduce the number of points you need to calculate.
• Step 5: Connect all points smoothly, respecting the end behavior and multiplicities. Remember that polynomial graphs have no sharp corners, gaps, or asymptotes. The graph should flow naturally from one zero to the next.

Common Graphing Mistakes

• Forgetting to check multiplicity can lead to drawing an incorrect graph shape at a zero. Always determine whether to cross or turn at each intercept before sketching.
• Misidentifying end behavior by looking at all terms instead of just the leading term is a common error. Focus on the highest-degree term only when predicting end behavior.
• Not plotting enough points between zeros can result in a misleading graph shape. Higher-degree polynomials often require more than just intercepts to capture their full curvature.
• Ignoring symmetry when it exists can waste time. If all exponents are even, the graph is symmetric about the y-axis; if all are odd, it’s symmetric about the origin.
• Failing to check your sketch against known values can cause avoidable errors. Always verify at least one or two calculated points after drawing the graph.

Rational Functions

Definition and Characteristics

• A rational function is a function that can be expressed as the ratio of two polynomials: \( R(x) = \frac{P(x)}{Q(x)} \), where \(Q(x) \neq 0\). This means the denominator cannot be zero, as division by zero is undefined. The polynomials \(P(x)\) and \(Q(x)\) can be of any degree, which affects the function’s graph and asymptotic behavior.
• Rational functions can have discontinuities, which are points where the function is not defined. These can occur as vertical asymptotes (non-removable) or holes (removable discontinuities) depending on whether a factor in the denominator cancels with a factor in the numerator. Understanding this distinction is key to correctly graphing rational functions.
• Because rational functions involve division, their graphs can have multiple disconnected parts. Each part is bounded by asymptotes or influenced by holes in the domain. Recognizing these separate branches early helps in sketching accurate graphs.
• When the numerator’s degree is smaller than the denominator’s degree, the function’s end behavior approaches zero, creating a horizontal asymptote at \(y = 0\). This concept connects to limits and how polynomials grow at different rates for large \(|x|\).
• If the numerator and denominator share a common factor, the zero of that factor creates a hole in the graph rather than an asymptote. This is because the factor cancels out algebraically, but the original function remains undefined at that point.

Domain of Rational Functions

Finding Domain Restrictions

• To find the domain of a rational function, identify all x-values that make the denominator zero, since division by zero is undefined. For example, if \(Q(x) = x^2 - 9\), set \(x^2 - 9 = 0\) to find \(x = \pm 3\) as excluded values. This step is always the first thing to do when analyzing a rational function.
• The domain can be expressed in set notation or interval notation, excluding any x-values that make the denominator zero. For example, the domain of \(\frac{x+1}{x^2 - 9}\) is \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\). This notation clearly shows where the function is defined.
• If the numerator and denominator have a common factor, cancel it only after identifying restrictions from the original denominator. For example, \(\frac{(x-2)(x+1)}{(x-2)(x-3)}\) simplifies to \(\frac{x+1}{x-3}\), but \(x=2\) is still excluded from the domain because it was a restriction in the original form.
• Understanding domain restrictions is essential for correctly identifying holes in the graph. If a factor cancels, the restriction still exists but results in a hole instead of an asymptote. Failing to recognize this leads to incorrect graphing.
• When working with word problems, domain restrictions may also come from the context of the problem. For instance, if \(x\) represents time, negative values might be excluded even if they don’t make the denominator zero. Always consider both algebraic and contextual domain limits.

Vertical, Horizontal, and Slant Asymptotes

Vertical Asymptotes

• Vertical asymptotes occur where the denominator equals zero and the factor does not cancel with the numerator. For example, \(\frac{x+2}{x-3}\) has a vertical asymptote at \(x = 3\). The graph approaches but never crosses these lines, and the function value increases or decreases without bound near them.
• To find vertical asymptotes, factor the numerator and denominator fully, cancel any common factors, and then set the remaining denominator factors equal to zero. The solutions are the x-values of the vertical asymptotes. This method ensures you don’t confuse holes with asymptotes.
• The behavior of the graph near a vertical asymptote can differ on either side of the asymptote. Testing points just to the left and right of the asymptote shows whether the function approaches \(+\infty\) or \(-\infty\) on each side. This helps in sketching accurate graphs.
• In applied settings, vertical asymptotes can represent limits or constraints—such as a tank’s water level approaching a maximum capacity but never reaching it. Understanding their real-world meaning makes them easier to interpret in modeling problems.
• Common mistake: treating a hole as a vertical asymptote. Remember, if a factor cancels, the graph will have a hole at that x-value instead of an infinite spike.

Horizontal Asymptotes

• Horizontal asymptotes describe the y-value the function approaches as \(x\) becomes very large or very small. They depend on the degrees of the numerator (\(n\)) and denominator (\(m\)). This is found by comparing how fast each polynomial grows.
• If \(n < m\), the horizontal asymptote is \(y = 0\). For example, \(\frac{x+1}{x^2+4}\) has \(n=1\), \(m=2\), so it flattens toward zero as \(|x|\) grows. This is because the denominator’s growth rate dominates.
• If \(n = m\), the horizontal asymptote is \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\). For example, \(\frac{2x^3 + 1}{5x^3 - 7}\) has \(y = \frac{2}{5}\). This happens because the highest-degree terms dominate both numerator and denominator.
• If \(n > m\), there is no horizontal asymptote; instead, the function’s end behavior is unbounded. This leads to the possibility of a slant (oblique) asymptote or even higher-degree asymptotes.
• Horizontal asymptotes can be crossed by the graph for rational functions, unlike vertical asymptotes. They describe end behavior, not a strict boundary for all x-values.

Slant (Oblique) Asymptotes

• Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. For example, \(\frac{x^2+3x+2}{x+1}\) has a slant asymptote because \(n=2\) and \(m=1\). This creates a diagonal line that the graph approaches at infinity.
• To find a slant asymptote, perform polynomial long division (or synthetic division) on the numerator by the denominator. The quotient (without the remainder) is the equation of the slant asymptote. The remainder becomes insignificant as \(|x|\) grows.
• Slant asymptotes describe the tilted end behavior of the function when neither a horizontal asymptote nor bounded end behavior applies. This happens when the numerator’s growth outpaces the denominator’s by exactly one degree.
• Just like horizontal asymptotes, slant asymptotes can be crossed in the middle of the graph. The asymptote only dictates behavior for large \(|x|\) values, not local deviations.
• In more advanced cases, when \(n > m+1\), the graph can have a polynomial asymptote of higher degree instead of just a slant. This is less common in basic AP Precalculus but worth noting for completeness.

Holes in Rational Functions

Identifying and Plotting Holes

• Holes, or removable discontinuities, occur in rational functions when a factor in the numerator and denominator is identical and cancels out. For example, in \(\frac{(x-2)(x+3)}{(x-2)(x-5)}\), the \((x-2)\) factor cancels, indicating a hole at \(x = 2\). The cancellation removes the infinite discontinuity but does not make the function defined at that x-value.
• To find the location of a hole, first factor both the numerator and denominator completely. Cancel common factors, and set the canceled factor equal to zero to find the x-coordinate of the hole. This ensures you do not mistake a hole for a vertical asymptote.
• Once you know the x-coordinate of the hole, find its y-coordinate by substituting it into the simplified function (after cancellation). For the example above, substituting \(x = 2\) into \(\frac{x+3}{x-5}\) gives \(y = \frac{5}{-3}\), so the hole is at \((2, -\frac{5}{3})\).
• Graphically, a hole is shown as an open circle on the curve at the point where the function would be defined if the factor were not canceled. This open point tells the viewer that the function value is undefined there even though the curve passes near it.
• Common mistake: forgetting that a canceled factor still affects the domain. Even if the factor is gone in the simplified equation, the original restriction from the denominator remains. This is why the hole exists in the first place.

Graphing Rational Functions

Step-by-Step Graphing Process

• Step 1: Identify the domain by finding values that make the denominator zero. Mark these as potential vertical asymptotes or holes depending on whether the factor cancels with the numerator. This step sets the foundation for the rest of the graph.
• Step 2: Determine vertical asymptotes by setting the remaining denominator factors equal to zero after canceling common factors. These vertical lines will be drawn on the graph and are approached but never crossed by the curve.
• Step 3: Identify holes by locating canceled factors and finding their coordinates. Mark these with open circles when sketching to indicate the missing point on the curve.
• Step 4: Find horizontal or slant asymptotes by comparing the degrees of the numerator and denominator. This will tell you the long-term behavior of the graph and how it approaches certain y-values for large \(|x|\).
• Step 5: Find the y-intercept by substituting \(x = 0\) into the original equation, and find x-intercepts by setting the numerator equal to zero (ignoring canceled factors). These intercepts help anchor your sketch and give clear reference points.
• Step 6: Plot additional points between intercepts and near asymptotes to understand the curve’s exact shape. Test points close to each vertical asymptote from both sides to determine whether the function approaches \(+\infty\) or \(-\infty\).
• Step 7: Draw the curve smoothly, respecting asymptotes, holes, and intercepts. Remember that rational function graphs are always continuous within each section of their domain, so there should be no breaks except at asymptotes and holes.

Common Graphing Errors

• Confusing a hole with a vertical asymptote. Always check for canceled factors before deciding it’s an asymptote.
• Forgetting that rational function graphs can cross horizontal asymptotes in the middle. Asymptotes only describe end behavior, not all x-values.
• Neglecting to check behavior near vertical asymptotes from both sides. This often leads to incorrect placement of the curve’s rise or fall.
• Not plotting enough points to capture the curvature between asymptotes. Some sections may curve differently than expected without extra verification points.
• Forgetting to mark holes explicitly on the graph, which can mislead viewers into thinking the function is defined there.

Modeling with Polynomial and Rational Functions

Using Polynomial Models

• Polynomial models are useful when the relationship between variables is smooth and predictable, without sudden breaks or asymptotes. For example, projectile motion height can be modeled by a quadratic, and certain revenue functions can be modeled by cubic polynomials. These models allow for interpolation and prediction within a given data range.
• To create a polynomial model, determine the degree needed based on the number of turning points or intercepts the data suggests. Then, use known points to write equations for the coefficients, either through substitution or regression tools. This process ensures the model fits the data closely.
• Check the model’s end behavior to ensure it makes sense in context. For example, if the model predicts negative population for large time values, it might only be valid over a short range. This is a crucial part of identifying limitations.
• Polynomial models can overfit data if the degree is too high. Overfitting makes the model less reliable for predictions outside the data range. Always balance the degree with simplicity and interpretability.
• Use polynomial models when you expect the dependent variable to eventually grow faster than any constant or logarithmic rate, as this matches the behavior of higher-degree polynomials.

Using Rational Models

• Rational models are best for situations where rates, proportions, or diminishing returns occur. For example, rational functions can model speed as a function of time in motion problems with resistance, or the concentration of a drug in the bloodstream over time. These scenarios often involve asymptotic behavior.
• To build a rational model, identify variables that cause restrictions in the domain and match them to denominator factors. Then, determine the numerator based on how the output behaves for small and large values of the input. This ensures the model reflects both local and long-term behavior.
• Interpret vertical asymptotes as constraints or points of breakdown in the system. For example, a machine’s output might approach infinity as input speed nears a critical limit, indicating potential failure or inefficiency.
• Horizontal or slant asymptotes in rational models represent long-term limits. For instance, a company’s profit might approach a maximum possible value as production increases, reflecting capacity limitations.
• Rational models should be tested for both asymptotic behavior and fit with real-world data. A model might mathematically fit but still be invalid if it predicts impossible values within the intended range of use.

Identifying Assumptions and Limitations of Function Models

Recognizing Model Assumptions

• Every mathematical model makes assumptions about the relationship between variables. For example, a quadratic model for projectile motion assumes no air resistance and a constant acceleration due to gravity. These assumptions are often unstated, but they are critical for knowing when the model applies.
• Polynomial models assume the dependent variable changes smoothly without sudden jumps or breaks. If the real-world data shows abrupt changes, the polynomial model may not be appropriate. Recognizing this prevents misapplication of the model.
• Rational models often assume certain constraints that cause asymptotic behavior, such as limited capacity or diminishing returns. These assumptions must reflect the real-world system being modeled; otherwise, predictions will be inaccurate.
• Checking assumptions involves comparing model predictions to known behavior. If the model’s end behavior does not match the expected long-term trend, it is likely that one or more assumptions are wrong. This is why end behavior analysis is important before trusting predictions.
• Models may also assume that all variables and parameters remain constant over time. If conditions change, the model’s accuracy decreases, and it may need to be adjusted or replaced with a different type of function.

Understanding Model Limitations

• All models are approximations and have limits to their usefulness. A polynomial model might fit data well over a short range but fail to predict accurately outside that range. This is called poor extrapolation, and it is a common problem in real-world applications.
• Rational models may predict unrealistic values near vertical asymptotes if those asymptotes do not correspond to actual physical limits. In such cases, the model might overestimate or underestimate the true output of the system.
• Limitations can also come from overfitting the data. If the degree of a polynomial is too high, it may capture random noise rather than the true underlying trend, leading to unreliable predictions. Always aim for the simplest model that fits the data well.
• Sometimes, models are limited by the precision of the data used to create them. Small measurement errors can lead to large differences in predicted values when extrapolated far beyond the data range. This is especially true for higher-degree polynomials.
• Understanding limitations helps in deciding whether to trust a model’s predictions or use it only for descriptive purposes. In many cases, a model is good for identifying trends but not for precise long-term forecasting.

Common Misconceptions

Polynomial Functions

• Misconception: All polynomial graphs look like simple parabolas or cubic curves. In reality, higher-degree polynomials can have multiple turns, varying steepness, and complex shapes depending on coefficients and multiplicities of zeros.
• Misconception: The number of real zeros always equals the degree of the polynomial. The degree gives the maximum number of real zeros, but some zeros may be complex and not visible on the graph.
• Misconception: You can determine a polynomial’s end behavior by looking at all its terms. Only the leading term (highest degree) determines end behavior for large \(|x|\), so focus on that term when analyzing.
• Misconception: Multiplicity only affects whether a graph crosses or touches the x-axis. In fact, higher multiplicities also flatten the graph near the zero, which affects the curvature and slope around that point.
• Misconception: Factoring is always easy for polynomials. In reality, some polynomials require advanced techniques or numerical methods to find zeros, especially if they do not factor nicely over the rationals.

Rational Functions

• Misconception: Vertical asymptotes are the only type of discontinuity. Holes (removable discontinuities) are equally important and must be identified separately by checking for common factors in numerator and denominator.
• Misconception: Horizontal asymptotes cannot be crossed. Rational functions can cross horizontal asymptotes in the middle of the graph; the asymptote only describes behavior as \(|x|\) becomes large.
• Misconception: Slant asymptotes are just tilted vertical lines. They are actually oblique lines that represent the function’s end behavior when the numerator’s degree is exactly one greater than the denominator’s.
• Misconception: Domain restrictions disappear if a factor cancels. Even if a factor is removed algebraically, the original x-value remains excluded from the domain, creating a hole in the graph.
• Misconception: Rational models are always accurate near asymptotes. In many real-world cases, values predicted near vertical asymptotes are meaningless because the model’s assumptions fail close to those points.

Practice Problem 1: Polynomial Functions

Problem

• Given the polynomial function \(f(x) = 2x^4 - 5x^3 - 8x^2 + 20x\):
  1. Factor the polynomial completely.
  2. Identify all real zeros and their multiplicities.
  3. Determine the end behavior of the graph.
  4. State the number of turning points the graph can have at most.

Solution

• Step 1- Factor Completely: First, factor out the GCF of \(x\): \(f(x) = x(2x^3 - 5x^2 - 8x + 20)\). Group terms: \((2x^3 - 5x^2) - (8x - 20)\). Factor each group: \(x^2(2x - 5) - 4(2x - 5)\). Factor out the common binomial: \((2x - 5)(x^2 - 4)\). Factor \(x^2 - 4\) into \((x - 2)(x + 2)\). Final factored form: \[ f(x) = x(2x - 5)(x - 2)(x + 2) \]
• Step 2- Identify Zeros & Multiplicities: Zeros: \(x = 0\) (mult. 1), \(x = \frac{5}{2}\) (mult. 1), \(x = 2\) (mult. 1), \(x = -2\) (mult. 1). All multiplicities are 1, so the graph will cross the x-axis at each zero.
• Step 3- End Behavior: The leading term is \(2x^4\), which has even degree and a positive coefficient. Therefore: as \(x \to \pm\infty\), \(f(x) \to +\infty\). Both ends of the graph rise.
• Step 4- Turning Points: A degree-4 polynomial can have at most \(4 - 1 = 3\) turning points. The actual number depends on the specific coefficients, but the maximum possible is 3.

Practice Problem 2: Rational Functions

Problem

• Given the rational function \(R(x) = \frac{x^2 - 9x + 14}{x^2 - 5x + 6}\):
  1. Identify the domain.
  2. State the location of any holes.
  3. Determine all vertical and horizontal asymptotes.
  4. Find the x- and y-intercepts.

Solution

• Step 1- Domain: Factor denominator: \(x^2 - 5x + 6 = (x - 2)(x - 3)\). The denominator cannot be zero, so \(x \neq 2\) and \(x \neq 3\). Domain: \((-\infty, 2) \cup (2, 3) \cup (3, \infty)\).
• Step 2- Holes: Factor numerator: \(x^2 - 9x + 14 = (x - 7)(x - 2)\). The \((x - 2)\) factor cancels, so \(x = 2\) is a hole. To find the hole’s y-value, plug \(x = 2\) into the simplified function \(\frac{x - 7}{x - 3}\): \(\frac{2 - 7}{2 - 3} = \frac{-5}{-1} = 5\). Hole at \((2, 5)\).
• Step 3- Vertical & Horizontal Asymptotes: After canceling, the denominator is \(x - 3\), so the vertical asymptote is \(x = 3\). Degrees of numerator and denominator are both 1, so horizontal asymptote is \(y = \frac{\text{LC of numerator}}{\text{LC of denominator}} = \frac{1}{1} = 1\).
• Step 4- Intercepts: x-intercept: set numerator of simplified function \((x - 7) = 0\) → \(x = 7\), so \((7, 0)\). y-intercept: substitute \(x = 0\) into original equation: \(\frac{14}{6} = \frac{7}{3}\), so \((0, \frac{7}{3})\).