Chapters 2.1-2.3

Q10. Use the graph of the function f shown to answer parts (a)-(n).

Background

Topic: Analyzing Functions Using Graphs

This question tests your ability to interpret a function's graph to find function values, intercepts, domain, range, and to answer questions about the behavior of the function at specific points. These are foundational skills in Precalculus, especially in understanding how functions behave visually and algebraically.

Key Terms and Formulas:

• Function Value: $f(a)$ is the y-value of the function at $x = a$.

• x-intercept: Points where $f(x) = 0$ (the graph crosses the x-axis).

• y-intercept: The point where $x = 0$.

• Domain: All possible $x$-values for which $f(x)$ is defined.

• Range: All possible $y$-values that $f(x)$ can take.

Step-by-Step Guidance

1. Finding function values: To find $f(a)$ for a given $x = a$, locate $a$ on the x-axis and read the corresponding y-value from the graph.

2. Determining sign (positive/negative): If the point is above the x-axis, $f(a)$ is positive; if below, $f(a)$ is negative.

3. Finding x-intercepts: Identify all points where the graph crosses the x-axis. These are the $x$-values where $f(x) = 0$.

4. Finding y-intercepts: Find the point where the graph crosses the y-axis ($x = 0$). The y-coordinate at this point is the y-intercept.

5. Domain and Range: For the domain, look at the leftmost and rightmost $x$-values on the graph. For the range, look at the lowest and highest $y$-values the graph reaches.

6. Special values: For questions like "For what $x$ is $f(x) = k$?", find where the graph crosses the horizontal line $y = k$ and read the corresponding $x$-values.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $f(-7) = -4$, $f(-2) = 4$

• (b) $f(6) = 4$, $f(0) = -2$

• (c) $f(2)$ is negative

• (d) $f(-3)$ is positive

• (e) $f(x) = 0$ at $x = -6, -1, 4$

• (f) $f(x) > 0$ for $-6 < x < -1$ and $4 < x \leq 6$

• (g) Domain: $-7 \leq x \leq 6$

• (h) Range: $-4 \leq y \leq 6$

• (i) x-intercepts: $x = -6, -1, 4$

• (j) y-intercept: $y = -2$

• (k) The line $y = 1$ intersects the graph 3 times

• (l) The line $x = 5$ intersects the graph 1 time

• (m) $f(x) = -4$ at $x = -7, 2$

• (n) $f(x) = 6$ at $x = -3$

Each answer was found by carefully reading the graph at the specified points or intervals.