Question

Graph the polynomial by transforming an appropriate graph of the form y=x^n. Show clearly all x- and y-intercepts. P(x)=−3(x+2)^5+96

Polynomial graphs
Graph the polynomial by transforming an appropriate graph of the form $$\displaystyle{y}={x}^{{n}}$$. Show clearly all x- and y-intercepts. $$\displaystyle{P}{\left({x}\right)}=−{3}{\left({x}+{2}\right)}^{{5}}+{96}$$

2021-05-10

The parent function of P(x) = $$\displaystyle—{3}{\left({x}+{2}\right)}^{{5}}+{96}$$ is $$\displaystyle{y}={x}^{{5}}$$, which passes through the points (—2,—32), (—1,—1), (0,0), (1,1), and (2,32).
$$\displaystyle{P}{\left({x}\right)}=—{3}{\left({c}+{2}\right)}^{{5}}+{96}$$ is the graph of $$y = x^5$$ reflected across the x-axis, vertically stretched by 3, and translated left 2 units and up 96 units. Multiplying the y-coordinates of the points on $$y = x^5 by -3$$ to reflect them across the x-axis and vertically stretch them by 3 gives the points (-2,96), (-1,3), (0,0), (1, -3), and (2,-96). Translating the points left 2 units and up 96 units then gives the points (-4, 192), (-3,99), (-2,96), (-1,93), and (0,0).
The x-intercept of P(x) is when $$P(x) = 0$$ which we already know is at (0,0). Note that P(x) can have only one x-intercept since $$y = x^{5}$$ has only one x-intercept.
The y-intercept is when $$x = 0$$ which we know is at (0,0).
Plot the points and then connect them with a smooth curve. Label the coordinates at the interceps in your graph.
See the explanation for the graph. To make the graph, reflect the graph ot $$y = x^5$$ across the x-axis, vertically stretch it by 2, and then translate it left 2 mits and up 96 units. Label the intercept at (0,0).