9. explain how the graph of each given function is a transformation of the graph of y = x². (2 points)\na. y…

Explanation:

Step1: Analyze $y = x^{2}-5$

For a function $y = f(x)+k$, when $k=- 5$, the graph of $y = f(x)$ is shifted vertically. The graph of $y=x^{2}$ is shifted 5 units downwards since for each $x$-value, the $y$-value of $y = x^{2}-5$ is 5 less than that of $y = x^{2}$.

Step2: Analyze $y=-2x^{2}$

For a function $y = af(x)$, when $a=-2$. The negative sign reflects the graph of $y = f(x)$ about the $x$-axis, and $|a| = 2$ vertically stretches the graph of $y = x^{2}$ by a factor of 2. So the graph of $y=x^{2}$ is reflected about the $x$-axis and vertically stretched by a factor of 2.

Step3: Analyze $y=(x - 5)^{2}$

For a function $y=f(x - h)$, when $h = 5$. The graph of $y = f(x)$ is shifted horizontally. The graph of $y=x^{2}$ is shifted 5 units to the right because for the function $y=(x - 5)^{2}$, when $y$ has the same value as in $y=x^{2}$, the $x$-value in $y=(x - 5)^{2}$ is 5 units larger.

Step4: Analyze $y=(-3x)^{2}=9x^{2}$

For a function $y = f(bx)$, when $b=-3$ (or $|b| = 3$). First, $y=(-3x)^{2}=9x^{2}$. The graph of $y=x^{2}$ is horizontally compressed by a factor of $\frac{1}{3}$ (since for the same $y$-value, the $x$-value in $y = (-3x)^{2}$ is $\frac{1}{3}$ of the $x$-value in $y=x^{2}$). Also, since the coefficient of $x^{2}$ is 9, it is vertically stretched by a factor of 9.

Answer:

a. The graph of $y = x^{2}$ is shifted 5 units downwards. b. The graph of $y = x^{2}$ is reflected about the $x$-axis and vertically stretched by a factor of 2. c. The graph of $y = x^{2}$ is shifted 5 units to the right. d. The graph of $y = x^{2}$ is horizontally compressed by a factor of $\frac{1}{3}$ and vertically stretched by a factor of 9.