Precalculus Study Guide: Linear and Quadratic Functions, Models, and Inequalities

Q1. For each function, (i) determine the slope and y-intercept; (ii) graph the function using slope and y-intercept; (iii) determine the average rate of change; and (iv) determine whether the function is increasing, decreasing, or constant.

• (a) $f(x) = 3x + 5$

• (b) $f(x) = -4x + 2$

• (c) $f(x) = 5$

Background

Topic: Linear Functions and Their Properties

This question tests your understanding of linear functions, including how to identify their slope and y-intercept, graph them, calculate the average rate of change, and determine their behavior (increasing, decreasing, or constant).

Key Terms and Formulas

• Slope (m): The rate at which the function changes as x increases. For $f(x) = mx + b$, m is the slope.

• y-intercept (b): The value of y when $x = 0$.

• Average Rate of Change: For a linear function, this is the same as the slope.

• Increasing/Decreasing/Constant: Determined by the sign of the slope.

Step-by-Step Guidance

1. Identify the slope and y-intercept for each function by comparing to the standard form $f(x) = mx + b$.

2. Plot the y-intercept on the graph, then use the slope to find another point and draw the line.

3. Calculate the average rate of change (for linear functions, this is just the slope).

4. Determine if the function is increasing (positive slope), decreasing (negative slope), or constant (zero slope).

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Q2. Suppose $f(x) = 3x + 6$ and $g(x) = -x + 4$.

• (a) Solve $f(x) = 0$.

• (b) Solve $f(x) \geq 0$.

• (c) Solve $f(x) = g(x)$.

• (d) Solve $f(x) \leq g(x)$.

Background

Topic: Solving Linear Equations and Inequalities

This question asks you to solve equations and inequalities involving linear functions, including finding where two lines intersect and where one is above or below the other.

Key Terms and Formulas

• Solving $f(x) = 0$: Find x where the function crosses the x-axis.

• Solving $f(x) = g(x)$: Set the two functions equal and solve for x (intersection point).

• Solving inequalities: Find the set of x-values where the inequality holds.

Step-by-Step Guidance

1. For (a), set $3x + 6 = 0$ and solve for x.

2. For (b), solve $3x + 6 \geq 0$ for x.

3. For (c), set $3x + 6 = -x + 4$ and solve for x.

4. For (d), solve $3x + 6 \leq -x + 4$ for x.

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Q3. The cost, C, in dollars of a cellular phone plan is given by the function $C(x) = 0.30x + 7$, where x is the number of minutes used.

• (a) What is the cost of the plan if you talk for 150 minutes?

• (b) If the bill is $220, how many minutes were used?

• (c) What is the maximum number of minutes that can be used for $120?

Background

Topic: Linear Models and Applications

This question involves applying a linear model to real-world data, interpreting the slope and intercept in context, and solving for unknowns.

Key Terms and Formulas

• Linear Model: $C(x) = 0.30x + 7$

• Solving for C: Substitute x and compute.

• Solving for x: Rearrange the equation to solve for x when C is given.

Step-by-Step Guidance

1. For (a), substitute $x = 150$ into $C(x)$ and simplify.

2. For (b), set $C(x) = 220$ and solve for x.

3. For (c), set $C(x) = 120$ and solve for x.

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Q4. An object is propelled straight upward from a height of 6 feet with an initial velocity of 32 feet per second. The height at any time t is given by $s(t) = -16t^2 + 32t + 6$. Find the maximum height attained by the object.

Background

Topic: Quadratic Functions and Maximum/Minimum Values

This question tests your ability to analyze a quadratic function to find its vertex, which represents the maximum height in this context.

Key Terms and Formulas

• Vertex of a parabola: For $s(t) = at^2 + bt + c$, the vertex occurs at $t = -\frac{b}{2a}$.

• Maximum height: Substitute the vertex value of t back into $s(t)$.

Step-by-Step Guidance

1. Identify a, b, and c in the quadratic equation.

2. Calculate $t = -\frac{b}{2a}$ to find the time at which the maximum height occurs.

3. Substitute this value of t back into $s(t)$ to find the maximum height.

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Q5. A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. What dimensions will produce a maximum enclosed area?

Background

Topic: Quadratic Optimization (Applications of Quadratic Functions)

This question involves maximizing the area of a rectangle given a fixed perimeter, a classic optimization problem in precalculus.

Key Terms and Formulas

• Perimeter constraint: Express the total fencing in terms of the rectangle's dimensions.

• Area function: Write area as a function of one variable using the constraint.

• Maximum area: Find the vertex of the quadratic area function.

Step-by-Step Guidance

1. Let the dimensions of the corrals be x and y. Write the perimeter equation based on the fencing arrangement.

2. Express y in terms of x using the perimeter equation.

3. Write the area function $A(x)$ in terms of x only.

4. Find the value of x that maximizes $A(x)$ by finding the vertex.

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Q6. The revenue function for a new plasma television is given by $R(p) = 900p - 0.1p^2$. What price, p, should be charged to maximize revenue? What is the maximum revenue?

Background

Topic: Maximizing Quadratic Functions (Revenue Models)

This question asks you to find the price that maximizes revenue, which is the vertex of a downward-opening parabola.

Key Terms and Formulas

• Revenue function: $R(p) = 900p - 0.1p^2$

• Vertex formula: $p = -\frac{b}{2a}$ for $R(p) = ap^2 + bp + c$

Step-by-Step Guidance

1. Identify a and b in the quadratic revenue function.

2. Use the vertex formula to find the price p that maximizes revenue.

3. Substitute this value of p back into $R(p)$ to find the maximum revenue.

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Q7. The following data represent the percentage of the population in a certain country aged 40 or older who do not have a college degree for various age groups. Find a quadratic model that describes the relationship between age and percentage. Use the model to predict the percentage of 53-year-olds that do not have a college degree.

Age, x	25.4	32.4	41.3	51.3	54.5	26.1	29.8

Background

Topic: Quadratic Regression and Modeling

This question involves fitting a quadratic model to data and using the model to make predictions.

Key Terms and Formulas

• Quadratic regression: Finding a quadratic function $P(x) = ax^2 + bx + c$ that best fits the data.

• Prediction: Substitute the desired x-value into the model to estimate the percentage.

Step-by-Step Guidance

1. Use a calculator or software to perform quadratic regression on the data points to find the coefficients a, b, and c.

2. Write the quadratic model $P(x)$ using the coefficients found.

3. Substitute $x = 53$ into $P(x)$ to estimate the percentage for 53-year-olds.

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Q8. Solve each inequality:

• (a) $x^2 - x - 12 \leq 0$

• (b) $-2x^2 > -11x + 15$

• (c) $3x^2 + 6x > 45$

Background

Topic: Solving Quadratic Inequalities

This question tests your ability to solve inequalities involving quadratic expressions, which often requires factoring and analyzing sign charts.

Key Terms and Formulas

• Quadratic inequality: An inequality involving a quadratic expression.

• Factoring: Expressing the quadratic as a product of binomials (if possible).

• Sign chart: Used to determine where the quadratic is positive or negative.

Step-by-Step Guidance

1. Rewrite each inequality in standard form (all terms on one side, zero on the other).

2. Factor the quadratic expression if possible.

3. Find the roots (where the expression equals zero).

4. Use a sign chart or test intervals to determine where the inequality holds.

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Q9. Solve $f(x) > g(x)$ where $f(x) = 2x^2 + 2x + 3$ and $g(x) = x^2 - 2x + 8$.

Background

Topic: Quadratic Inequalities and Function Comparison

This question asks you to compare two quadratic functions and find the set of x-values where one is greater than the other.

Key Terms and Formulas

• Set up the inequality: $f(x) > g(x)$ becomes $2x^2 + 2x + 3 > x^2 - 2x + 8$.

• Combine like terms: Bring all terms to one side to form a single quadratic inequality.

• Solve the quadratic inequality: Factor or use the quadratic formula, then test intervals.

Step-by-Step Guidance

1. Subtract $g(x)$ from both sides to get a single quadratic inequality.

2. Simplify the expression and write in standard form.

3. Find the roots of the resulting quadratic equation.

4. Use a sign chart to determine where the inequality is satisfied.

Try solving on your own before revealing the answer!