Precalculus Practice Test 2 Study Guidance

Q1. What is the vertex of the graph of $f(x) = 2x^2 - 4x - 3$?

Background

Topic: Quadratic Functions and Their Graphs

This question tests your understanding of how to find the vertex of a quadratic function, which is a key feature of its graph.

Key Terms and Formulas

• Vertex: The point $(h, k)$ where the graph of a parabola reaches its maximum or minimum.

• Standard form: $f(x) = ax^2 + bx + c$

• Vertex formula: $h = -\frac{b}{2a}$, $k = f(h)$

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the quadratic: $a = 2$, $b = -4$, $c = -3$.

2. Use the formula $h = -\frac{b}{2a}$ to find the x-coordinate of the vertex.

3. Plug the value of $h$ back into $f(x)$ to find the y-coordinate $k$.

4. Write the vertex as $(h, k)$.

Try solving on your own before revealing the answer!

Final Answer: The vertex is $(1, -5)$

Using $h = -\frac{-4}{2 \cdot 2} = 1$, and $k = f(1) = 2(1)^2 - 4(1) - 3 = -5$, so the vertex is $(1, -5)$.

Q2. Write a quadratic function that expresses the area of a rectangular garden in terms of its width $w$, given a perimeter of 400 ft.

Background

Topic: Quadratic Modeling and Applications

This question tests your ability to model a real-world situation (area of a rectangle) using a quadratic function.

Key Terms and Formulas

• Perimeter of rectangle: $P = 2l + 2w$

• Area of rectangle: $A = l \cdot w$

Step-by-Step Guidance

1. Set up the perimeter equation: $2l + 2w = 400$.

2. Solve for $l$ in terms of $w$: $l = \frac{400 - 2w}{2}$.

3. Substitute $l$ into the area formula: $A = l \cdot w$.

4. Simplify to get a quadratic function in terms of $w$.

Try solving on your own before revealing the answer!

Final Answer: $A(w) = 200w - w^2$

Substituting $l = 200 - w$ into $A = l \cdot w$ gives $A(w) = w(200 - w) = 200w - w^2$.

Q3. Sketch the graph of a function $f$ such that $f(-3) = 1$, $f(0) = -5$, and $f$ decreases on $(-\infty, 4)$ and increases on $(4, \infty)$.

Background

Topic: Graphing Functions and Analyzing Intervals

This question tests your ability to interpret function values and describe increasing/decreasing behavior.

Key Terms

• Increasing/Decreasing Intervals: Where the function's output rises or falls as $x$ increases.

• Critical Point: Where the function changes from decreasing to increasing (often a minimum).

Step-by-Step Guidance

1. Plot the points $(-3, 1)$ and $(0, -5)$ on your graph.

2. Note that the function decreases until $x = 4$, then increases after $x = 4$.

3. Identify $x = 4$ as a likely minimum point.

4. Sketch a curve that passes through the given points and has a minimum at $x = 4$.

Try solving on your own before revealing the answer!

Final Answer: The graph is a parabola opening upward with a minimum at $x = 4$.

The function passes through $(-3, 1)$ and $(0, -5)$, decreases until $x = 4$, then increases. The minimum is at $(4, f(4))$.

Q4. (a) Given the graph of $g(x)$, draw a rough sketch of its inverse, labeling at least one point with coordinates. (b) State the domain and range of the function $h(x)$ whose graph is shown.

Background

Topic: Inverse Functions and Domain/Range

This question tests your understanding of how to sketch the inverse of a function and how to determine domain and range from a graph.

Key Terms

• Inverse Function: The function that 'undoes' the original function, reflected over $y = x$.

• Domain: Set of possible input values ($x$).

• Range: Set of possible output values ($y$).

Step-by-Step Guidance

1. For the inverse, reflect the graph of $g(x)$ over the line $y = x$.

2. Choose a point on $g(x)$, such as $(a, b)$, and plot $(b, a)$ for the inverse.

3. For $h(x)$, examine the graph to determine the lowest and highest $x$ and $y$ values.

4. Write the domain and range based on the graph's extent.

Try solving on your own before revealing the answer!

Final Answer:

(a) The inverse of $g(x)$ is a curve reflected over $y = x$, e.g., if $g(2) = 0$, then the inverse passes through $(0, 2)$.

(b) The domain of $h(x)$ is $[0, \infty)$ and the range is $[0, \infty)$, based on the graph.

Q5. The daily profit for a manufacturer is given by $P(x) = -8x^2 + 5000x - 8000$, where $P$ is in hundreds of dollars and $x$ is the number of units produced. (a) How many units should be produced each day to yield a maximum profit? (b) State or show how you would find the maximum profit (do not compute it).

Background

Topic: Quadratic Optimization

This question tests your ability to find the maximum of a quadratic function, which models profit.

Key Terms and Formulas

• Maximum of a parabola: For $ax^2 + bx + c$, maximum occurs at $x = -\frac{b}{2a}$ if $a < 0$.

Step-by-Step Guidance

1. Identify $a = -8$, $b = 5000$ in the profit function.

2. Use $x = -\frac{b}{2a}$ to find the number of units for maximum profit.

3. To find the maximum profit, substitute this $x$ value into $P(x)$.

Try solving on your own before revealing the answer!

Final Answer:

(a) Maximum profit occurs at $x = \frac{5000}{16} = 312.5$ units.

(b) To find the maximum profit, plug $x = 312.5$ into $P(x)$: $P(312.5)$.

Q6. Let $m(x) = x^3 - 2x$. (a) Find all zeroes of $m$ and describe the right-hand and left-hand behavior of the graph. (b) Determine algebraically whether $m$ is even, odd, or neither. (c) Draw a rough sketch of the graph of $m$, labeling the x-intercepts.

Background

Topic: Polynomial Functions and Symmetry

This question tests your ability to find roots, analyze end behavior, and determine function symmetry.

Key Terms and Formulas

• Zeroes: Values of $x$ where $m(x) = 0$.

• Even/Odd Function: Even if $m(-x) = m(x)$, odd if $m(-x) = -m(x)$.

• End Behavior: How the function behaves as $x \to \pm\infty$.

Step-by-Step Guidance

1. Set $m(x) = 0$ and solve for $x$.

2. Factor or use the cubic formula to find roots.

3. Check $m(-x)$ and compare to $m(x)$ and $-m(x)$ to determine symmetry.

4. Describe end behavior: As $x \to \infty$, $m(x) \to \infty$; as $x \to -\infty$, $m(x) \to -\infty$.

5. Sketch the graph, labeling x-intercepts.

Try solving on your own before revealing the answer!

Final Answer:

(a) Zeroes are $x = 0$, $x = \sqrt{2}$, $x = -\sqrt{2}$.

(b) $m(x)$ is an odd function because $m(-x) = -m(x)$.

(c) The graph passes through $(0,0)$, $(\sqrt{2},0)$, and $(-\sqrt{2},0)$, with end behavior as described.

Q7. (a) Write the equation for a function whose graph is the result of first reflecting $y = x$ about the x-axis, then shifting it 4 units right and 2 units down. (b) State the range and y-intercept of this function.

Background

Topic: Transformations of Functions

This question tests your understanding of how to apply reflections and translations to a function.

Key Terms and Formulas

• Reflection about x-axis: $y = -f(x)$

• Horizontal shift: $y = f(x - h)$

• Vertical shift: $y = f(x) + k$

Step-by-Step Guidance

1. Start with $y = x$.

2. Reflect about x-axis: $y = -x$.

3. Shift 4 units right: $y = -(x - 4)$.

4. Shift 2 units down: $y = -(x - 4) - 2$.

5. Find the y-intercept by plugging $x = 0$.

6. State the range based on the transformed function.

Try solving on your own before revealing the answer!

Final Answer:

(a) The equation is $y = -(x - 4) - 2 = -x + 4 - 2 = -x + 2$.

(b) The range is $(-\infty, \infty)$ and the y-intercept is $y = 2$.

Q8. Let $f(x) = \frac{3}{x - 9}$. (a) What is the y-intercept of $f$? (b) What is the equation of the asymptote of the graph of $f$? (c) Sketch the graph of $f$, labeling intercepts and asymptotes.

Background

Topic: Rational Functions and Asymptotes

This question tests your ability to find intercepts and asymptotes of rational functions.

Key Terms and Formulas

• Y-intercept: Value of $f(0)$.

• Vertical asymptote: Where denominator is zero.

• Horizontal asymptote: Behavior as $x \to \pm\infty$.

Step-by-Step Guidance

1. Find y-intercept by plugging $x = 0$ into $f(x)$.

2. Set denominator to zero to find vertical asymptote.

3. Analyze end behavior for horizontal asymptote.

4. Sketch the graph, labeling intercepts and asymptotes.

Try solving on your own before revealing the answer!

Final Answer:

(a) Y-intercept is $f(0) = -\frac{1}{3}$.

(b) Vertical asymptote: $x = 9$. Horizontal asymptote: $y = 0$.

(c) The graph has a vertical asymptote at $x = 9$, horizontal at $y = 0$, and y-intercept at $(0, -\frac{1}{3})$.

Q9. Use composition of functions to determine whether $p(x) = x + 3$ and $q(x) = x - 2$ are inverses.

Background

Topic: Function Composition and Inverses

This question tests your ability to use composition to check if two functions are inverses.

Key Terms and Formulas

• Inverse functions: $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the domain.

• Composition: $(f \circ g)(x) = f(g(x))$

Step-by-Step Guidance

1. Compute $p(q(x))$ and $q(p(x))$.

2. Check if both compositions equal $x$.

Try solving on your own before revealing the answer!

Final Answer:

$p(q(x)) = (x - 2) + 3 = x + 1$; $q(p(x)) = (x + 3) - 2 = x + 1$. Since neither composition equals $x$, $p$ and $q$ are not inverses.

Q10. (a) Let $f(x) = \frac{2x^2 + 8}{x - 2}$ and $g(x) = x^2 - 4$. Find $\frac{f(x)}{g(x)}$, writing your answer in simplest form, and state its domain. (b) Evaluate $\log_4 1 + \ln 9 - 2$. (c) Determine the equations of all asymptotes, vertical and horizontal, of the graph of $y = \frac{2x^2 - 12}{x^2 - 9}$.

Background

Topic: Rational Functions, Logarithms, and Asymptotes

This question tests your ability to simplify rational expressions, evaluate logarithmic expressions, and find asymptotes.

Key Terms and Formulas

• Rational function: $\frac{f(x)}{g(x)}$

• Domain: Values of $x$ for which the function is defined.

• Logarithm properties: $\log_b 1 = 0$, $\ln a$ is the natural log.

• Asymptotes: Vertical where denominator is zero, horizontal from end behavior.

Step-by-Step Guidance

1. Write $\frac{f(x)}{g(x)} = \frac{2x^2 + 8}{(x - 2)(x^2 - 4)}$.

2. Simplify numerator and denominator, factor where possible.

3. State domain: Exclude values where denominator is zero.

4. For logs, use properties: $\log_4 1 = 0$, $\ln 9 = 2\ln 3$.

5. For asymptotes, set denominator to zero for vertical, compare degrees for horizontal.

Try solving on your own before revealing the answer!

Final Answer:

(a) $\frac{f(x)}{g(x)} = \frac{2x^2 + 8}{x^3 - 4x + 8}$, domain excludes $x = 2$ and $x = -2$.

(b) $\log_4 1 = 0$, $\ln 9 = 2.197$, so $0 + 2.197 - 2 = 0.197$.

(c) Vertical asymptotes: $x = 3$, $x = -3$. Horizontal asymptote: $y = 2$.