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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 17
Chapter 4, Problem 17

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−4)2−1

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Identify the vertex of the parabola from the function f(x) = (x - 4)^2 - 1. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Here, h = 4 and k = -1, so the vertex is (4, -1).
Write the equation of the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex, so its equation is x = h. For this function, the axis of symmetry is x = 4.
Find the y-intercept by evaluating f(0). Substitute x = 0 into the function: f(0) = (0 - 4)^2 - 1 = 16 - 1 = 15. So, the y-intercept is (0, 15).
Find the x-intercepts by setting f(x) = 0 and solving for x. Set (x - 4)^2 - 1 = 0, then solve the equation: (x - 4)^2 = 1, which gives two solutions for x.
Determine the domain and range of the function. The domain of any quadratic function is all real numbers, so domain: (-∞, ∞). The range depends on the vertex and the direction of the parabola. Since the parabola opens upward and the vertex is the minimum point at y = -1, the range is [ -1, ∞ ).

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Key Concepts

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

Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the parabola changes direction, representing either its maximum or minimum value. It is given by the coordinates (h, k) in the vertex form f(x) = a(x - h)^2 + k. Understanding the vertex helps in graphing the parabola and determining its key features.
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Vertex Form

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is x = h, where h is the x-coordinate of the vertex. This line is crucial for graphing and analyzing the symmetry properties of quadratic functions.
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Properties of Parabolas

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since the parabola extends infinitely left and right. The range depends on the vertex; for a parabola opening upwards, the range is all y-values greater than or equal to the vertex's y-coordinate. Understanding domain and range helps describe the function's output values.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

f(x)=(x−3)2f(x)=(x−3)^2

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 5x≤2−3x25x≤2−3x^2

Textbook Question

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x3−2x2−11x+12=0

Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as z and the sum of y and w.

Textbook Question

Divide using synthetic division. (2x2+x−10)÷(x−2)

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Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→−2+, f(x)→x\(\to\)-2^{+},\(\text{ }\)f(x)\(\to\) __

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