Question

\(f(x)=8 x^2-6\); horizontal stretch by a factor of 2 and a translation 2 units up, followed by a reflection in the \(y\)-axis

Short answer

The transformed function, after horizontal stretch by a factor of 2, translation 2 units up, and reflection in the \(y\)-axis, is given by \(f_3(x)= 2x^2 - 4\)

Step-by-step solution

Horizontal Stretch

Applying the horizontal stretch by a factor of 2 to the function, we plug \(x/2\) into \(f(x)\): \(f_1(x)= f\left(\frac{x}{2}\right) = 8\left(\frac{x}{2}\right)^2-6\), Simplifying yields \(f_1(x)= 2x^2 - 6\)

Translation Upwards

Next, we apply the upward shift of 2 units, so we add 2 to \(f_1(x)\): \(f_2(x)= f_1(x) + 2\). Substituting \(f_1(x)\) we get: \(f_2(x)= 2x^2 - 6 + 2\). Hence, \(f_2(x)= 2x^2 - 4\)

Reflection in the \(y\)-axis

Finally, we reflect the function obtained in step 2 in the \(y\)-axis. We achieve this by replacing \(x\) with \(-x\) in the function \(f_2(x)\) that results in \(f_3(x)= f_2(-x) = 2(-x)^2 - 4\). So, \(f_3(x)= 2x^2 - 4\)

Key concepts

Horizontal Stretch

A horizontal stretch involves stretching a function along the horizontal axis. This transformation occurs when you multiply the input value, or the "x" value, of the function by a factor. In our example, we applied a horizontal stretch to the function \(f(x) = 8x^2 - 6\) by a factor of 2.

To achieve this, you replace \(x\) with \(\frac{x}{2}\). This is because a horizontal stretch by a factor \(a\) means you substitute \(x\) with \(\frac{x}{a}\). So, our function becomes:

• \(f_1(x) = f\left(\frac{x}{2}\right) = 8\left(\frac{x}{2}\right)^2 - 6\)

Simplifying this gives us \(f_1(x) = 2x^2 - 6\). The function is now stretched by a factor of 2, making it wider without changing its vertical size or shape.

Vertical Translation

Vertical translation is simply shifting the entire graph of the function up or down the vertical axis. This transformation does not alter the shape or orientation of the graph; it just slides it vertically. In our example from the function \(f_1(x) = 2x^2 - 6\), a translation of 2 units up was performed.

To execute this shift, you add 2 to the whole function, which results in the new function:

• \(f_2(x) = 2x^2 - 6 + 2 = 2x^2 - 4\)

The graph of \(f_2(x)\) is now the same shape as \(f_1(x)\) but positioned 2 units higher on the graph, making the curve's position adjusted but still having the same width and shape.

Reflection in the Y-Axis

Reflecting a function in the y-axis means flipping it over the vertical y-axis, creating a mirror image of the original graph. To reflect a graph in the y-axis, the x-values in the equation of the function are replaced by their opposites. In our step-by-step example, this is applied to the function \(f_2(x) = 2x^2 - 4\).

We replace \(x\) with \(-x\) to get the reflected function:

• \(f_3(x) = f_2(-x) = 2(-x)^2 - 4\)

Interestingly, since squaring \(-x\) results in the same as squaring \(x\), the reflection does not change the equation for the parabola, resulting in \(f_3(x) = 2x^2 - 4\). This means the final function looks the same after reflection because its original and reflected forms coincide due to the symmetrical nature of parabolas.