QuestionWhat happens to the values of $f(x)=x^{3}+5$ compared to $y=x^{3}$ ? How does this change affect the graph?

Studdy Solution

STEP 1

Assumptions1. The original function is $y = x^{3}$ . The new function is $f(x) = x^{3} +5$
3. We are asked to determine the effect of adding5 to the original function on the function's values and its graph.

STEP 2

First, let's understand the effect of adding5 to the function on its values.In the original function, $y = x^{}$ , the output $y$ is the cube of the input $x$ .
In the new function, $f(x) = x^{} +5$ , the output is the cube of the input $x$ , plus5.
So, for any given $x$ , the value of $f(x)$ will be5 more than the value of $y$ .

STEP 3

Let's visualize this with an example.
For $x =2$ , in the original function, $y =2^{3} =8$ .
In the new function, $f(2) =2^{3} +5 =8 +5 =13$ .
So, the value of the function at $x =2$ is increased by5.

STEP 4

Now, let's understand the effect of adding to the function on its graph.
The graph of the function $y = x^{3}$ is a curve that passes through the origin (0,0) and increases for positive $x$ and decreases for negative $x$ .
When we add to the function to get $f(x) = x^{3} +$ , this has the effect of shifting the entire graph of $y = x^{3}$ upward by units.

STEP 5

To visualize this, consider the point (2,8) on the graph of $y = x^{3}$ .
After adding5 to the function, the corresponding point on the graph of $f(x) = x^{3} +5$ is (2,13), which is5 units above the point (2,8).
So, the graph of $f(x) = x^{3} +5$ is the graph of $y = x^{3}$ shifted upward by5 units.
In conclusion, adding5 to the function $y = x^{3}$ increases the function's values by5 for any given $x$ , and shifts the graph of the function upward by5 units.

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