Question

How can I create a quadratic function that results in a horizontal stretch by a factor of 4?

Answer 1

To create a quadratic function that results in a horizontal stretch by a factor of 4, you can start with the standard form of a quadratic function, which is:

\[
f(x) = ax^2 + bx + c
\]

A horizontal stretch by a factor of \( k \) can be achieved by replacing \( x \) with \( \frac{x}{k} \) in the function. In your case, since you want a stretch by a factor of 4, you will replace \( x \) with \( \frac{x}{4} \).

Let's assume we start with the basic quadratic function \( f(x) = x^2 \). To stretch it horizontally by a factor of 4, we modify it as follows:

\[
f(x) = \left(\frac{x}{4}\right)^2
\]

Now let's simplify that:

\[
f(x) = \frac{x^2}{16}
\]

So, a quadratic function that results in a horizontal stretch by a factor of 4 can be expressed as:

\[
f(x) = \frac{1}{16}x^2
\]

You may also see the function expressed in vertex or standard form (e.g., \( f(x) = a(x - h)^2 + k \)), but the concept remains the same: replacing \( x \) with \( \frac{x}{4} \) in the function accomplishes the desired horizontal stretch.

If you have a function of the form \( f(x) = a(x - h)^2 + k \), applying the horizontal stretch by a factor of 4 will result in:

\[
f(x) = a\left(\frac{x}{4} - h\right)^2 + k
\]

This will stretch the graph horizontally while preserving its vertical characteristics.