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 11
Chapter 4, Problem 11

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5

Verified step by step guidance
1
Identify the form of the quadratic function. The given function is in vertex form: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
Rewrite the function to match the vertex form template. The function is \(f(x) = -2(x + 1)^2 + 5\), which can be seen as \(f(x) = -2(x - (-1))^2 + 5\).
From the expression, determine the vertex coordinates by comparing to \(f(x) = a(x - h)^2 + k\). Here, \(h = -1\) and \(k = 5\).
Therefore, the vertex of the parabola is at the point \((-1, 5)\).
Remember that the coefficient \(a = -2\) indicates the parabola opens downward, but this does not affect the vertex coordinates.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex directly from the equation without further calculations.
Recommended video:
08:07
Vertex Form

Effect of Parameters a, h, and k on the Parabola

In the vertex form, 'a' determines the parabola's direction and width: if a is negative, the parabola opens downward; if positive, upward. The values 'h' and 'k' shift the parabola horizontally and vertically, respectively, locating the vertex at (h, k).
Recommended video:
7:12
Parabolas as Conic Sections Example 1

Finding the Vertex Coordinates

To find the vertex coordinates from a quadratic in vertex form, identify h and k from the expression f(x) = a(x - h)^2 + k. Note the sign inside the parentheses is opposite to the vertex's x-coordinate. The vertex is then (h, k), which gives the maximum or minimum point of the parabola.
Recommended video:
Guided course
02:16
Graphs and Coordinates - Example
Related Practice
Textbook Question

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).]

f(x)=x6−6x4+9x2 f(x) = x^6 -6x^4 + 9x^2

1576
views
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.

As x→1−, f(x)→x\(\to\)1^{-},\(\text{ }\)f(x)\(\to\)_{_{}}______

865
views
Textbook Question

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

534
views
Textbook Question

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).] f(x)=−x3+x2+2xf(x) = -x^3 + x^2 + 2x

a.b. c. d.

1202
views
Textbook Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3−2x2−11x+12

1081
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. 3x2+10x−8≤0

499
views