Problem 41 - Algebra 2

Question

REASONING How do you change the equation of the parent function \(y=x^{2}\) to shift the graph to the right?

Step-by-Step Solution

Answer

Change the equation to \(y = (x - h)^2\), where \(h\) is the number of units shifted to the right.

Step 1: Understand the Horizontal Shift

To shift a graph horizontally, we change the variable inside the function. Specifically, for a rightward shift, we subtract a positive number from the variable inside the function.

Step 2: Apply the Horizontal Shift to the Parent Function

The parent function given is \(y = x^2\). To shift this graph to the right by \(h\) units, we replace \(x\) with \(x - h\). This modifies the equation to \(y = (x - h)^2\).

Step 3: Generalize the Solution

The transformation results in the equation \(y = (x - h)^2\), where \(h\) represents the number of units shifted to the right. This equation produces a graph of a parabola identical in shape to \(y = x^2\), but moved \(h\) units to the right along the x-axis.

Key Concepts

Horizontal Shift

When graphing functions like a parabola, understanding horizontal shifts is crucial. A horizontal shift involves moving the entire graph left or right along the x-axis without altering its shape. In the case of a quadratic function, such as the parent function \( y = x^2 \), a horizontal shift is achieved by adjusting the variable inside the function's argument.To perform a rightward shift of a graph, you modify the function by subtracting a number from the variable \( x \). For instance, replacing \( x \) with \( x - h \) in the equation causes the graph to move \( h \) units to the right. The new equation, \( y = (x - h)^2 \), represents this shifted parabola.It's important to remember:

• Shifting right: subtract from \( x \)
• Each unit subtracted moves the graph one unit to the right

This fundamental understanding of horizontal shifts helps in easily predicting the graph's new position.

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable \( x \) is 2. The standard form of a quadratic function is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).One of the most basic forms of a quadratic function is the parent function \( y = x^2 \). In this case:

• \( a = 1 \), \( b = 0 \), \( c = 0 \)
• The parabola opens upwards, forming a 'U' shape.
• The vertex of the parabola is at the origin, \((0,0)\).

Understanding quadratic functions is key to grasping the effects of transformations like horizontal shifts. Changes to the function's coefficients or variables inside the equation will modify the graph's position, size, or orientation.

Parent Function

A parent function serves as a template for a family of functions. For quadratic functions, the parent function is \( y = x^2 \). It represents the simplest form of a quadratic equation with no transformations.Key characteristics of the parent function \( y = x^2 \) include:

• Graph is a parabola opening upwards.
• Symmetrical about the y-axis.
• The vertex is at the point \((0,0)\).
• No horizontal or vertical shifts applied.

The concept of parent functions is fundamental in understanding how transforms like shifting, stretching, or reflecting can generate new graphs. In practice, being able to identify the parent function allows you to quickly interpret changes brought by transformations.

Graphing Parabolas

Graphing parabolas involves plotting the smooth, U-shaped curve of a quadratic function. The parabola's shape and position can be altered through various transformations. For the parent quadratic function \( y = x^2 \), the graph can be modified by transformations such as horizontal and vertical shifts, reflection, and scaling.To graph a parabola like \( y = (x - h)^2 \):

• Identify the vertex, which in this case is \((h, 0)\), due to the horizontal shift.
• Plot the vertex as a starting point.
• Because \( a \) is positive in \( y = x^2 \), the parabola opens upwards.
• Mark additional points on either side of the vertex to define the parabola's width and exact shape.

Graphing a parabola with transformations can visually demonstrate the effects of shifting and other modifications. This process builds a deeper understanding of quadratic functions in a graphical context.