Question

Consider the graph of the function f. a) Find the domain, range, and zeros of the function. b) write an equation for the function f. (In vertex form, standard form, or intercept form)c) compare the graph of f to the graph of g(x) = x^2.

Answer 1

Given the graph;

(a) The domain of a function is the set of values for which the function is real and defined. Thus, the domain D is;

`\begin{gathered} (-\infty,\infty) \\ D:All\text{ }real\text{ }numbers \end{gathered}`

The range is;

`y\leq8`

The zeros of the function are the points y=0;

`x=1,x=5`

(b) The equation of a parabola in vertex form is;

`\begin{gathered} y=a(x-h)^2+k \\ Where\text{ }(h,k)\text{ is }the\text{ }vertex; \\ and\text{ }given\text{ }(1,0) \\ \\ 0=a(1-3)^2+8 \\ \\ -8=4a \\ \\ a=-2 \\ \\ \end{gathered}`

Thus, the equation is;

`y=-2(x-3)^2+8`

(c) Using the graph below;

The graph of g(x) has its intercept at (0,0).

The transformation goes as;

Vertical stretch 2units, reflection over the x-axis, horizontal shift to the the right 3 units, vertical shift up 8 units

Using (5,0);

`\begin{gathered} y=a(x-h)^2+k \\ \\ 0=a(5-3)^2+8 \\ 4a=-8 \\ \\ a=-2 \end{gathered}`