Question

The graph of \(y=x^{2}\) is shown in the standard \((x, y)\) coordinate plane below. For which of the following equations is the graph of the parabola shifted 3 units to the right and 2 units down? F. \(y=(x+3)^{2}+2\) G. \(y=(x+3)^{2}-2\) H. \(y=(x-2)^{2}+3\) J. \(y=(x-3)^{2}+2\) K. \(y=(x-3)^{2}-2\)

Short answer

Answer: K. \(y=(x-3)^{2}-2\)

Step-by-step solution

Shift 3 Units to the Right

To shift the given graph 3 units to the right, we need to replace \(x\) with \((x-3)\). The new equation for the graph shifted 3 units to the right will be \(y=(x-3)^2\).

Shift 2 Units Down

To further shift the graph 2 units down, we need to subtract 2 from the equation in Step 1. The final equation representing the transformation will be \(y=(x-3)^2 - 2\). After the transformations, the correct equation of the parabola is given by: \(y = (x-3)^2 - 2\)

Identify the Correct Answer Choice

By comparing our derived equation to the given options, we find that the correct answer is: K. \(y=(x-3)^{2}-2\)

Key concepts

Quadratic Transformations

Understanding quadratic transformations is critical when graphing parabolas. A basic quadratic function is represented by the equation \(y = x^2\). However, we can apply transformations to shift or stretch this graph on the coordinate plane. When the graph shifts to the right or left, it's a horizontal translation, and when it moves up or down, it's a vertical translation.

To shift the graph horizontally, we adjust the x-value within the parentheses. For instance, \(y = (x - h)^2\) shifts the parabola h units to the right if h is positive, and h units to the left if h is negative. To move the graph vertically, we modify the value at the end of the equation. Adding a positive value \(k\) to the end, \(y = x^2 + k\), shifts the parabola up, while subtracting \(k\) as in \(y = x^2 - k\), shifts it down.

In our exercise, we needed to apply both horizontal and vertical translations to the parabola. The correct transformation involved shifting the graph 3 units to the right and 2 units down, which can be achieved by the equation \(y = (x - 3)^2 - 2\).

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is determined by a pair of numbers, known as coordinates. These numbers correspond to the distances from two intersecting lines, the x-axis (horizontal) and y-axis (vertical).

When graphing the equation of a parabola, like \(y = x^2\), each x-value is squared to give the corresponding y-value, forming a U-shaped curve that opens upwards. When we graph transformations of this basic equation, we can see how the curve moves in relation to these axes. For example, if an equation has \(x - 3\), the entire graph moves 3 units to the right, which means every point on the parabola is 3 units away from where it would be on the graph of \(y = x^2\).

In the exercise, identifying how the graph moves in relation to the coordinate axes was key to choosing the correct equation. Empowering landscape shots with a sense of nearby presence, while the graph of \(y = x^2\) sits at the origin, the correct transformed equation shows the parabola shifted according to the given translations.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging, adding, subtracting, multiplying, or dividing parts of algebraic expressions to simplify them or to solve equations. It is indispensable in solving problems involving quadratic equations and transformations.

In our textbook exercise, algebraic manipulation came into play when we adjusted the quadratic equation to achieve the desired translation of the graph. Firstly, we substituted \(x\) with \(x - 3\) to represent a 3-unit shift to the right. Then, we algebraically manipulated the equation by subtracting 2 to account for the 2-unit downward shift, resulting in the equation \(y = (x - 3)^2 - 2\).

Being comfortable with these manipulations can help students understand the effects of transformations on the graph and then correctly match the transformed equation to the changes described in a problem. It is a foundational skill that extends beyond graphing parabolas, assisting in a wide variety of algebraic tasks.