Skip to main content
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 3
Chapter 4, Problem 3

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=−x2+2x+3f(x) = -x^2 +2x + 3

Verified step by step guidance
1
Identify the quadratic function: \(f(x) = -x^2 + 2x + 3\). Notice it is a parabola opening downward because the coefficient of \(x^2\) is negative.
Find the vertex using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\). Here, \(a = -1\) and \(b = 2\), so calculate \(x = -\frac{2}{2 \times (-1)}\).
Substitute the x-coordinate of the vertex back into the function to find the y-coordinate: \(f\left(-\frac{b}{2a}\right) = -\left(-\frac{b}{2a}\right)^2 + 2\left(-\frac{b}{2a}\right) + 3\).
Find the y-intercept by evaluating \(f(0)\), which gives the point where the graph crosses the y-axis.
Find the x-intercepts by solving the quadratic equation \(-x^2 + 2x + 3 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -1\), \(b = 2\), and \(c = 3\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
12m
Was this helpful?

Key Concepts

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

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex's x-coordinate is found using -b/(2a). The vertex helps in sketching the graph and determining the range.
Recommended video:
08:07
Vertex Form

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is x = -b/(2a). This line is crucial for graphing and understanding the parabola's symmetry.
Recommended video:
07:42
Properties of Parabolas

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the parabola's direction: if it opens downward (a < 0), the range is all values less than or equal to the vertex's y-coordinate; if upward (a > 0), it is all values greater than or equal to the vertex's y-coordinate.
Recommended video:
4:22
Domain & Range of Transformed Functions
Related Practice
Textbook Question

Find the domain of each rational function. g(x)=3x2/(x−5)(x+4)

4
views
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. (x+1)(x−7)≤0

5
views
Textbook Question

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−x2+19x+6

6
views
Textbook Question

Divide using long division. State the quotient, and the remainder, r(x). (x3+5x2+7x+2)÷(x+2)

5
views
Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies inversely as x. y = 12 when x = 5. Find y when x = 2.

Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=6x7+πx5+2/3 x