Question

find the function rule for g(x).

1. the function f(x) = x². the graph of g(x) is f(x) translated to the left 6 units and down 5 units. what is the function rule for g(x)?

a. g(x) = (x + 6)² - 5 c. g(x) = (x - 6)² + 5
b. g(x) = (x - 5)² + 6 d. g(x) = (x + 5)² - 6

1. which rule defines the function in the graph shown?

graph of a piecewise function with three parts: a horizontal segment, a line with positive slope, and a line with negative slope, on a coordinate grid
a. f(x) = \

$$\begin{cases} -6, & -10 < x \\leq 2 \\\\ -2x, & 4 < x < 16 \\\\ x + 2, & -6 \\leq x < -14 \\end{cases}$$

c. f(x) = \

$$\begin{cases} -6, & -10 < x \\leq 2 \\\\ 2x, & 2 < x < 8 \\\\ -x + 2, & 8 \\leq x < 16 \\end{cases}$$

b. f(x) = \

$$\begin{cases} -6, & -10 \\leq x < 2 \\\\ -2x, & 4 < x < 16 \\\\ x + 2, & -6 \\leq x < -14 \\end{cases}$$

d. f(x) = \

$$\begin{cases} -6, & -10 \\leq x < 2 \\\\ 2x, & 2 \\leq x \\leq 8 \\\\ -x + 2, & 8 < x \\leq 16 \\end{cases}$$

Explanation:

Question 8

Step 1: Recall translation rules

For a function \( y = f(x) \), a horizontal translation left by \( h \) units is \( y = f(x + h) \), and a vertical translation down by \( k \) units is \( y = f(x) - k \).

Step 2: Apply to \( f(x)=x^2 \)

Left 6 units: replace \( x \) with \( x + 6 \), so \( f(x + 6)=(x + 6)^2 \). Down 5 units: subtract 5, so \( g(x)=(x + 6)^2 - 5 \).

Question 9

Step 1: Analyze the horizontal segment

The horizontal part (constant \( -6 \)): From the graph, it's for \( -10\leq x<2 \) (matches the domain in option d).

Step 2: Analyze the increasing line

The line with positive slope: Let's find its equation. Using points (2, 4) and (8, 16) (from the graph), slope \( m=\frac{16 - 4}{8 - 2}=\frac{12}{6}=2 \), equation \( y - 4 = 2(x - 2)\Rightarrow y = 2x \), domain \( 2\leq x\leq8 \) (matches option d).

Step 3: Analyze the decreasing line

The line with negative slope: Using points (8, -6) and (16, -14) (from the graph), slope \( m=\frac{-14 - (-6)}{16 - 8}=\frac{-8}{8}=-1 \), equation \( y - (-6)=-1(x - 8)\Rightarrow y=-x + 2 \), domain \( 8

Answer:

a. \( g(x) = (x + 6)^2 - 5 \)