Problem 32 - Algebra 2 and Trigonometry

Question

In \(28-33,\) without graphing the parabola, describe the translation, reflection, and \(/\) or scaling that must be applied to \(y=x^{2}\) to obtain the graph of each given function. $$ \mathrm{f}(x)=-x^{2}+x+2 $$

Step-by-Step Solution

Answer

Reflect over x-axis, move right by \(\frac{1}{2}\) units, move up by \(\frac{9}{4}\) units.

Step 1: Identify the base function

The base function is the standard parabola given by \( y = x^2 \). Our task is to find how this base function has been transformed into \( f(x) = -x^2 + x + 2 \).

Step 2: Recognize the reflection over the x-axis

The term \(-x^2\) indicates a reflection. Since the \(x^2\) term is negated, the parabola is reflected over the x-axis. This transformation changes the orientation from opening upwards to opening downwards.

Step 3: Identify horizontal translation

To check for horizontal translation, complete the square: \( f(x) = -x^2 + x + 2 \) can be rewritten by first factoring out a \(-1\) from the quadratic part: \(-((x^2 - x) - 2)\). Now focus on the expression \(x^2 - x\). Completing the square, \(x^2 - x = (x - \frac{1}{2})^2 - \frac{1}{4}\). So, \(-((x - \frac{1}{2})^2 - \frac{1}{4} - 2) = -(x - \frac{1}{2})^2 + \frac{9}{4}\), indicating a horizontal translation right by \(\frac{1}{2}\) units.

Step 4: Identify vertical translation

The constant term after completing the square was \(\frac{9}{4}\). This part of the completed square form, along with \(x\) terms, indicates an upward shift of \(\frac{9}{4}\) units from the x-axis.

Step 5: Combine all transformations

The function \(f(x) = -x^2 + x + 2\) is generated from \(y = x^2\) by first reflecting over the x-axis, translating to the right by \(\frac{1}{2}\) unit, and then vertically upwards by \(\frac{9}{4}\) units. There is no uniform scaling (i.e., a multiplied factor other than \(-1\)) of \( y = x^2 \).

Key Concepts

Reflection Over X-Axis

When you reflect a parabola over the x-axis, you flip its orientation. The standard parabola, represented by the function \(y = x^2\), opens upwards. In the function \(f(x) = -x^2 + x + 2\), the term \(-x^2\) signifies that our parabola has been flipped. This transformation changes it from opening upwards to opening downwards. Think of it like looking in a mirror placed along the x-axis, where the top becomes the bottom. This reflection over the x-axis is a fundamental transformation that alters the direction but not the shape of the parabola.

Horizontal Translation

Horizontal translation involves shifting the parabola left or right along the x-axis. To observe this in \(f(x) = -x^2 + x + 2\), we should rewrite the equation in a form that reveals any horizontal shift. By completing the square on the quadratic expression \(x^2 - x\), it transforms into \((x - \frac{1}{2})^2 - \frac{1}{4}\). The key here is the \(x - \frac{1}{2}\) term, which tells us the vertex moves right by \(\frac{1}{2}\) units. This is because a transformation \((x - h)\) denotes a horizontal shift to the right by \(h\) units.

Vertical Translation

Vertical translation moves the parabola up or down along the y-axis. In our function, this effect is observed through the constant term derived after completing the square. Given, \(f(x) = -(x - \frac{1}{2})^2 + \frac{9}{4}\), this form reveals a movement upward. The constant \(\frac{9}{4}\) tells us that the entire parabola is shifted up by \(\frac{9}{4}\) units. Essentially, compared to the standard base parabola \(y = x^2\), the whole graph lifts by this precise amount due to the vertical translation.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a form that easily shows transformations. For the function \(f(x) = -x^2 + x + 2\), it helps clarify both horizontal and vertical translations. First, focus on \(x^2 - x\). We know half of the linear coefficient \((-\frac{1}{2})\) squared \(\big(\frac{1}{4}\big)\) is added and subtracted inside the square, forming \((x - \frac{1}{2})^2\). This reveals the vertex or the turning point of the parabola, helping us identify shift directions precisely. Completing the square not only simplifies understanding of transformations but also aids in visualizing the graph's movements.