Question

Suppose the graph of a polynomial function P(x) has an x-intercept at (c, 0). Write 4-6 sentences analyzing whether (c, 0) is also an x-intercept of the graphs that result from applying single transformations to P(x). For each transformation, elaborate on whether (c, 0) is always, sometimes, or never an x-intercept of the transformed graph, explaining your reasoning. For any transformations for which (c, 0) is only sometimes an x-intercept of the transformed graph, design specific functions to illustrate how the other x-intercepts of the original graph determine whether (c, 0) is an x-intercept of the transformed graph.

Answer 1

For example:

Let:

P(x) = x² - 1

P(x) has an x-intercept at x=1 or x=-1 because:

x² - 1 = 0

x² = 1

√(x²) = √(1)

x = ±1

When we talk about transformations of functions, we apply three fundamental processes:

Translation, Scaling, Reflection

Translation: allows us to move a function to the left, right, up or down.

Scaling: allows us to expand or narrow a function

Reflection: allows us to reflect the function with respect to the x or y axis.

Let's apply a scaling:

For example:

P(x) = 2 (x² - 1)

As you can see, the polynomial function compresses horizontally by a factor of 1/2, however its intercept remains the same, because:

2 (x² - 1) = 0

Solving for x:

x = ±1

Let's apply a translation:

For example:

P(x) = (x² - 1) - 4

In this case the x-intercept will be different, because:

(x² - 1) - 4 = 0

Solving for x:

x² = 5

√(x²) = √(5)

x = ±√(5) = ± 2.236