Math

Question Analyze the effect of changing y=x3y=x^{3} to f(x)=(x+1)3f(x)=(x+1)^{3}. Identify the values most affected and describe the impact on the graph.

Studdy Solution

STEP 1

Assumptions
1. The original function is y=x3y = x^3.
2. The new function is f(x)=(x+1)3f(x) = (x + 1)^3.
3. We want to determine which values are most directly affected by the change from the original function to the new function.
4. We want to understand the effect of the change on the graph of the function.

STEP 2

To understand which values are most directly affected, we compare the two functions y=x3y = x^3 and f(x)=(x+1)3f(x) = (x + 1)^3.

STEP 3

Notice that for any value of xx, the new function f(x)f(x) is equivalent to evaluating the original function at x+1x + 1.
f(x)=(x+1)3=yy=(x+1)3f(x) = (x + 1)^3 = y|_{y = (x+1)^3}

STEP 4

This means that the graph of f(x)f(x) is the graph of y=x3y = x^3 shifted to the left by 1 unit.

STEP 5

The values most directly affected by the change are the xx-values of the function, since each xx-value in the new function corresponds to an xx-value that is one less than in the original function.

STEP 6

The effect on the graph would be a horizontal shift. Since we are adding one inside the function (to the xx value before it is cubed), the graph will shift to the left by 1 unit.

STEP 7

To visualize this, we can consider a few points on the original graph of y=x3y = x^3 and see where they would be on the graph of f(x)=(x+1)3f(x) = (x + 1)^3.

STEP 8

For example, the point (0,0)(0, 0) on the graph of y=x3y = x^3 would move to (1,0)(-1, 0) on the graph of f(x)f(x).

STEP 9

Similarly, the point (1,1)(1, 1) on the graph of y=x3y = x^3 would move to (0,1)(0, 1) on the graph of f(x)f(x).

STEP 10

This shift does not affect the shape of the graph; it only affects its position on the coordinate plane.

STEP 11

Therefore, the values most directly affected by the change are the xx-values, and the effect on the graph is a horizontal shift to the left by 1 unit.
Select an answer for the values most directly affected by the change: The xx-values.
Select an answer for the effect it would have on the graph: A horizontal shift to the left by 1 unit.

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