Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

4. Polynomial Functions

Quadratic Functions

College Algebra

4. Polynomial Functions

Quadratic Functions

Previous problemNext problem

1:28 minutes

Problem 11

Textbook Question

In Exercises 9–16, 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

Identify the standard form of a quadratic function, which is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Compare the given function \( f(x) = -2(x+1)^2 + 5 \) with the standard form to identify the values of \(a\), \(h\), and \(k\).

Notice that \(a = -2\), \(h = -1\), and \(k = 5\) from the function \( f(x) = -2(x+1)^2 + 5 \).

Recognize that the vertex \((h, k)\) of the parabola is \((-1, 5)\).

Conclude that the coordinates of the vertex for the given quadratic function are \((-1, 5)\).

Verified video answer for a similar problem:

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

Video duration:

Play a video:

Was this helpful?

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 can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the standard form and vertex form of quadratic functions is essential for analyzing their properties.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For a quadratic function in vertex form, f(x) = a(x-h)^2 + k, the vertex is located at the point (h, k). Identifying the vertex is crucial for graphing the parabola and understanding its maximum or minimum value.

Transformation of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the given function f(x) = -2(x+1)^2 + 5, the expression (x+1) indicates a horizontal shift to the left by 1 unit, while the -2 indicates a vertical stretch and reflection over the x-axis. Recognizing these transformations helps in accurately determining the vertex and overall shape of the parabola.