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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 67
Chapter 3, Problem 67

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=x2+2

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1
Identify the type of function given. Since the function is \(f(x) = x^2 + 2\), it is a quadratic function, which graphs as a parabola.
Recognize the basic shape of the graph. The parent function is \(y = x^2\), which is a parabola opening upwards with its vertex at the origin \((0,0)\).
Determine the effect of the '+ 2' in the function. This constant shifts the entire graph of \(y = x^2\) vertically upward by 2 units.
Find the vertex of the parabola. Since the parent function's vertex is at \((0,0)\) and the graph is shifted up by 2, the new vertex is at \((0, 2)\).
Plot several points by choosing values for \(x\), calculating \(f(x) = x^2 + 2\), and then sketch the parabola opening upwards with vertex at \((0, 2)\).

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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 written as f(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and properties of quadratic functions is essential for graphing them accurately.
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Solving Quadratic Equations Using The Quadratic Formula

Vertex of a Parabola

The vertex is the highest or lowest point on the graph of a quadratic function, representing its maximum or minimum value. For f(x) = x² + 2, the vertex is at (0, 2), since the function is in vertex form with no linear term. Identifying the vertex helps in sketching the parabola correctly.
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Horizontal Parabolas

Graphing Techniques for Quadratic Functions

Graphing a quadratic involves plotting the vertex, finding additional points by substituting x-values, and using symmetry about the axis of symmetry (x = -b/2a). Recognizing shifts, such as the +2 in f(x) = x² + 2, indicates a vertical translation of the parabola upward by 2 units.
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Graphs of Logarithmic Functions
Related Practice
Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. ƒ(3t-2)

Textbook Question

For each function, find (a) ƒ(2) and (b) ƒ(-1).See Example 7. ƒ = {(2,5),(3,9),(-1,11),(5,3)}

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Textbook Question

For each function, find (a) ƒ(2) and (b) ƒ(-1). ƒ = {(-1,3),(4,7),(0,6),(2,2)}

Textbook Question

Find and interpret the average rate of change illustrated in each graph.

Textbook Question

For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (3, -5), parallel to y = 4

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Textbook Question

For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (-7, 4), perpendicular to y = 8