Linear Functions and Their Properties: Precalculus Study Guidance

Q1. Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.

Background

Topic: Linear Functions and Slope

This question tests your ability to recognize linear functions from a table of values and, if the function is linear, to calculate its slope.

Key Terms and Formulas:

• Linear function: A function of the form $f(x) = mx + b$.

• Slope ($m$): The rate of change between two points, calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Step-by-Step Guidance

1. Examine the differences between consecutive $x$ values and $y$ values in the table.

2. Calculate the change in $y$ for each step and check if it is constant.

3. If the change in $y$ is constant for each change in $x$, the function is linear.

4. Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ with any two points to find the slope.

Try solving on your own before revealing the answer!

Final Answer:

The function is linear because the change in $y$ for each unit increase in $x$ is constant (5).

The slope is $m = 5$.

This means for every increase of 1 in $x$, $y$ increases by 5.

Q2. Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.

Background

Topic: Linear Functions and Slope

This question tests your ability to analyze a table of values to determine linearity and calculate the slope if applicable.

Key Terms and Formulas:

• Linear function: $f(x) = mx + b$

• Slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step-by-Step Guidance

1. Check the differences between consecutive $x$ values and $y$ values.

2. Calculate the change in $y$ for each step and see if it is constant.

3. If the change in $y$ is not constant, the function is nonlinear. If it is constant, proceed to find the slope.

4. Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ with any two points to find the slope.

Try solving on your own before revealing the answer!

Final Answer:

The function is nonlinear because the change in $y$ is not constant for each unit increase in $x$.

For example, from $x=2$ to $x=3$, $y$ changes from $-4$ to $-3$ (change of $+1$), but from $x=3$ to $x=4$, $y$ changes from $-3$ to $-5$ (change of $-2$).

Q3. In parts (a)-(f), use the given figure.

Background

Topic: Solving Linear Equations and Inequalities Graphically

This question tests your ability to interpret a graph of a linear function and solve equations and inequalities based on the graph.

Key Terms and Formulas:

• Linear function: $y = f(x)$

• Solving $f(x) = k$: Find the $x$ value where the function equals $k$.

• Solving $f(x) > k$ or $f(x) \leq k$: Find the range of $x$ values where the function is above or below $k$.

Step-by-Step Guidance

1. Identify the points on the graph where $f(x)$ equals the given values (e.g., $f(x) = 70$ at $x = 40$).

2. For inequalities, determine the intervals on the $x$-axis where the function is above or below the specified value.

3. For $f(x) = 0$, find the $x$-intercept.

4. For compound inequalities, identify the range of $x$ values that satisfy both conditions.

Try solving on your own before revealing the answer!

Final Answer:

• (a) $f(x) = 70$ at $x = 40$

• (b) $f(x) = 119$ at $x = 96$

• (c) $f(x) = 0$ at $x = -40$

• (d) $f(x) > 70$ for $x > 40$

• (e) $f(x) \leq 119$ for $x \leq 96$

• (f) $0 < f(x) < 119$ for $-40 < x < 96$

Each answer is found by reading the graph and matching the $y$ values to the corresponding $x$ values.

Q4. In parts (a) and (b), use the given figure.

Background

Topic: Solving Linear Equations and Inequalities Graphically

This question tests your ability to solve equations and inequalities involving two linear functions using their graphs.

Key Terms and Formulas:

• Intersection point: Where $f(x) = g(x)$

• Inequality: $f(x) > g(x)$ means the $y$ value of $f(x)$ is greater than $g(x)$ for certain $x$ values.

Step-by-Step Guidance

1. Locate the intersection point of the two lines on the graph. This is where $f(x) = g(x)$.

2. For the inequality $f(x) > g(x)$, identify the region(s) on the $x$-axis where the $y$ value of $f(x)$ is above that of $g(x)$.

3. Use the coordinates of the intersection point to write the solution for the equation.

4. For the inequality, describe the interval(s) where $f(x)$ is greater than $g(x)$.

Try solving on your own before revealing the answer!

Final Answer:

• (a) $f(x) = g(x)$ at $x = -1$

• (b) $f(x) > g(x)$ for $x < -1$

The intersection point gives the solution to the equation, and the region where $f(x)$ is above $g(x)$ gives the solution to the inequality.