Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 60

Chapter 3, Problem 60

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. g(x) = x^2 + 2

Verified step by step guidance

Start by identifying the standard quadratic function, f(x) = x^2. This is a parabola that opens upwards with its vertex at the origin (0, 0). The graph is symmetric about the y-axis.

Next, observe the given function, g(x) = x^2 + 2. Notice that this is a transformation of the standard quadratic function. Specifically, the '+2' indicates a vertical shift.

To apply the vertical shift, move every point on the graph of f(x) = x^2 upward by 2 units. For example, the vertex of f(x) = x^2, which is at (0, 0), will now move to (0, 2).

Plot the new points after the vertical shift. For instance, if the original graph of f(x) = x^2 passes through (1, 1), the new graph will pass through (1, 3). Similarly, if it passes through (-1, 1), the new graph will pass through (-1, 3).

Finally, sketch the new graph of g(x) = x^2 + 2. It will have the same shape as the original parabola, but its vertex will be at (0, 2), and the entire graph will be shifted 2 units upward.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of quadratic functions is essential for analyzing their transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For quadratic functions, vertical shifts occur when a constant is added or subtracted from the function, such as in g(x) = x^2 + 2, which shifts the graph of f(x) = x^2 upward by 2 units. Recognizing these transformations helps in accurately graphing modified functions.

Standard Form of a Quadratic Function

The standard form of a quadratic function is f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants. This form allows for easy identification of the vertex and the direction of the parabola. In the context of transformations, understanding how changes in 'c' affect the graph's position is crucial for accurately representing the function g(x) = x^2 + 2.

Recommended video:

04:34

Converting Standard Form to Vertex Form

Related Practice

Textbook Question

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² - 6y -7=0

views

Textbook Question

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

views

Textbook Question

Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. ƒ^-1 (1)

Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

Textbook Question

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² − x + 2y + 1 = 0

views

Textbook Question

In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 4x+y-6=0

views