Problem 29 - 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. $$ \mathrm{f}(x)=x^{2}+2 x-2 $$

Step-by-Step Solution

Answer

The function is translated 1 unit left and 3 units down, with no reflection or scaling.

Step 1: Rewrite the Function in Vertex Form

To analyze the translation, reflection, and scaling, we first need to express the given quadratic function in the vertex form. The function is currently given as \( f(x) = x^2 + 2x - 2 \). To convert it to vertex form, complete the square on the terms involving \( x \):\[ f(x) = (x^2 + 2x + 1) - 1 - 2 \]. Simplifying gives \[ f(x) = (x+1)^2 - 3 \]. Now, we have the function in the form \( f(x) = (x +1)^2 - 3 \).

Step 2: Identify Translations

The function \( f(x) = (x +1)^2 - 3 \) can be compared to the standard vertex form \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. Here, \( h = -1 \) and \( k = -3 \). This means the graph of \( y = x^2 \) is translated 1 unit to the left and 3 units downward.

Step 3: Identify Reflection and Scaling

In the equation \( f(x) = (x+1)^2 - 3 \), the coefficient \( a = 1 \), which is positive and equal to 1. This indicates that there is no reflection (since \( a \) is positive) and no vertical scaling (since \( a \) is 1). The graph retains its original parabola shape.

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a useful way to express parabolas. It's represented as \( y = a(x-h)^2 + k \). Here, \((h, k)\) is the vertex of the parabola.
Converting a parabola equation to vertex form involves completing the square. This transformation helps in easily identifying the vertex and understanding the movement of the parabola in a coordinate plane.
In the provided example, \( f(x) = x^2 + 2x - 2 \) was converted to \( f(x) = (x+1)^2 - 3 \). This shows how the vertex form reveals the vertex. For our equation, the vertex is \((-1, -3)\).
Having the equation in this form allows you to easily spot how the parabola shifts from position, which brings us to the topic of function translation.

Translation of Functions

Translation is the shifting of a graph in any direction without changing its shape. For parabolas, translating depends on the changes in the \(h\) and \(k\) values in vertex form.
Translation can occur in various ways:

• A shift to the right or left signals a change in \(h\). If \(h\) is negative, the shift is left; if positive, the shift is right.
• A shift up or down modifies \(k\). If \(k\) is negative, the shift moves down, and if positive, it goes up.

In the example \(f(x) = (x+1)^2 - 3\):

• The \((x+1)\) implies a 1-unit shift left, because \(h = -1\).
• The \(-3\) indicates a 3-unit shift downward, thus \(k = -3\).

This translation demonstrates how the original position of the parabola \( y = x^2 \) moves to obtain the new function.

Reflection of Functions

Reflection flips the graph over a specific line, like the x-axis or y-axis. In parabolas, reflection is typically over the x-axis and is indicated by the coefficient \(a\) in the vertex form.
If the coefficient \(a\) is negative, it means the parabola opens downwards, creating a reflection over the x-axis. In contrast, a positive \(a\) means no reflection, and the parabola preserves the direction in which it opens.
In our function \(f(x) = (x+1)^2 - 3\), \(a = 1\) is positive, so there's no reflection that changes the direction of the parabola. It stays opening upward, just like the graph of \(y = x^2\).
Reflection seeks to mirror the function across the axis; however, it remains unaffected when \(a\) is positive.

Scaling of Functions

Scaling changes the width or the steepness of the parabola. This effect is controlled by the coefficient \(a\) in vertex form.
When \(\left|a\right| > 1 \), the parabola becomes narrower. When \(0 < \left|a\right| < 1 \), it becomes wider.
For our example function \(f(x) = (x+1)^2 - 3\), \(a = 1\). This value indicates there's no scaling, as the parabola maintains its original width and shape.
Understanding scaling is essential because it modifies how tightly the parabola "hugs" the y-axis. Therefore, recognizing that \(a = 1\) keeps the graph's usual appearance helps in analyzing how similar or different the transformation is from the base function \(y = x^2\).