Question

what is the inverse of $y = \frac{x - 9}{5}$?
use algebraic notation to show how to transform the red function to the black function.
$(x + 4)^2 - 7$
$(x - 4)^2 + 7$
$(x + 7)^2 - 4$
$(x - 7)^2 + 4$
where is the local maximum for the graph below? check the correct square.
on the graph below, mark all the interval(s) where the graph is increasing.

Explanation:

Step1: Swap x and y for inverse

$x = \frac{y - 9}{5}$

Step2: Multiply both sides by 5

$5x = y - 9$

Step3: Solve for y

$y = 5x + 9$
---

Step1: Identify base and target functions

Black: $y=x^2$, Red: $y=(x+4)^2-7$

Step2: Translate left 4 units

$y=(x+4)^2$

Step3: Translate down 7 units

$y=(x+4)^2-7$
---

Step1: Locate local maximum peak

Check highest curve point: left square near $x\approx0$
---

Step1: Identify increasing intervals

Check rising curve sections: $(-\infty, -1)$ and $(1, \infty)$

Answer:

1. Inverse function: $y = 5x + 9$
2. Transform black $y=x^2$: shift left 4 units, then down 7 units to get $y=(x+4)^2-7$
3. Local maximum: the top-left square (near the positive y-axis peak)
4. Increasing intervals: $(-\infty, -1)$ and $(1, \infty)$ (the leftmost and rightmost marked intervals)