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

3. Functions

Transformations

College Algebra

3. Functions

Transformations

Previous problemNext problem

1:56 minutes

Problem 62

Textbook Question

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. r(x) = -(x + 1)^2

Verified step by step guidance

Start by graphing the standard quadratic function \( f(x) = x^2 \). This is a parabola that opens upwards with its vertex at the origin (0,0).

Identify the transformation needed for \( r(x) = -(x + 1)^2 \). Notice that \( (x + 1) \) indicates a horizontal shift.

Shift the graph of \( f(x) = x^2 \) to the left by 1 unit to account for the \( (x + 1) \) term. This moves the vertex from (0,0) to (-1,0).

Apply the vertical reflection indicated by the negative sign in front of \( (x + 1)^2 \). This flips the parabola over the x-axis, making it open downwards.

The final graph of \( r(x) = -(x + 1)^2 \) is a downward-opening parabola with its vertex at (-1,0).

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 opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of the standard quadratic function, f(x) = x^2, is essential for applying transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function r(x) = -(x + 1)^2 involves a horizontal shift to the left by 1 unit and a reflection across the x-axis. Recognizing how these transformations affect the original graph is crucial for accurately graphing the new function.

Reflection

Reflection in the context of graphing refers to flipping the graph over a specific axis. In the function r(x) = -(x + 1)^2, the negative sign indicates a reflection over the x-axis. This means that all y-values of the original function are inverted, which alters the orientation of the parabola from opening upwards to opening downwards.