Question

7.1 i can create equations for absolute value functions and perform tran
score: 0/10 answered: 0/10
question 1
write the following function as both (a) an absolute value equation and (b) as a piecewise function.
a. absolute value equation:
y =
b. piecewise function (use equal on the lower x value domain):
f(x) =
submit question

Explanation:

Step1: Identify vertex of the graph

The vertex (lowest point) of the absolute value graph is at $(3, -3)$.

Step2: Use vertex form of absolute value function

The vertex form is $y = a|x - h| + k$, where $(h,k)$ is the vertex. Substitute $h=3$, $k=-3$:
$y = a|x - 3| - 3$

Step3: Solve for slope $a$

Use the y-intercept $(0,5)$ (from the graph where $x=0$, $y=5$):
$5 = a|0 - 3| - 3$
$5 = 3a - 3$
$3a = 8$
$a = \frac{8}{3}$
So the absolute value equation is $y = \frac{8}{3}|x - 3| - 3$.

Step4: Find equations for piecewise function

For $x \leq 3$ (left side), the slope is $-\frac{8}{3}$. Use point-slope form $y - y_1 = m(x - x_1)$ with $(3,-3)$:
$y - (-3) = -\frac{8}{3}(x - 3)$
$y + 3 = -\frac{8}{3}x + 8$
$y = -\frac{8}{3}x + 5$

For $x > 3$ (right side), the slope is $\frac{8}{3}$. Use point-slope form:
$y - (-3) = \frac{8}{3}(x - 3)$
$y + 3 = \frac{8}{3}x - 8$
$y = \frac{8}{3}x - 11$

Answer:

A. Absolute Value Equation:
$y = \frac{8}{3}|x - 3| - 3$

B. Piecewise function:

$$ f(x) =
LATEXBLOCK0
$$