Question

use transformations of the standard cubic function, f(x) = x^3, to graph the function h(x) = 2(x - 3)^3 + 5. use the graphing tool to graph the function. click to enlarge graph (for any answer boxes shown with the grapher, type an exact answer.)

Explanation:

Step1: Identify horizontal shift

The function $h(x)=2(x - 3)^{3}+5$ is of the form $y = a(x - h)^{3}+k$. Here $h = 3$. For the standard cubic function $y=x^{3}$, replacing $x$ with $x - 3$ shifts the graph 3 units to the right.

Step2: Identify vertical stretch

The coefficient $a = 2$. This vertically stretches the graph of $y=(x - 3)^{3}$ by a factor of 2. That is, for each $y$-value of $y=(x - 3)^{3}$, the $y$-value of $y = 2(x - 3)^{3}$ is twice as large.

Step3: Identify vertical shift

The value $k = 5$. This shifts the graph of $y = 2(x - 3)^{3}$ 5 units up.

To graph the function, first graph the standard cubic function $y=x^{3}$ which has points such as $(-2,-8),(-1,-1),(0,0),(1,1),(2,8)$.
After the horizontal - shift (replace $x$ with $x - 3$), the points become $(1,-8),(2,-1),(3,0),(4,1),(5,8)$.
After the vertical - stretch (multiply $y$ - values by 2), the points become $(1,-16),(2,-2),(3,0),(4,2),(5,16)$.
After the vertical - shift (add 5 to $y$ - values), the points become $(1,-11),(2,3),(3,5),(4,7),(5,21)$. Plot these points and draw a smooth curve to get the graph of $h(x)=2(x - 3)^{3}+5$.

Answer:

Graph the function $h(x)=2(x - 3)^{3}+5$ by first shifting the graph of $y = x^{3}$ 3 units to the right, then vertically stretching it by a factor of 2, and finally shifting it 5 units up.