7.17.4 Transforming the Parent Function

Algebra 17.17.4 Transforming the Parent Function

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Activity

A graph of three parabolas on a coordinate grid. The x-axis extends from negative 5 to 5 with a scale of 1. The y-axis extends from 0 to 60 with a scale of 10.

Do you see two "eyes" and a smiling "mouth" on the graph? The three arcs on the graph all represent quadratic functions that were initially defined by $y = x^{2}$ , but whose equations were later modified.

Write an equation to represent the smile in the smiley face.

Compare your answer:

The mouth can be given by $y = x^{2} + 10$ because it is the graph of $y = x^{2}$ shifted 10 units upward.

Write an equation to represent the left eye in the smiley face.

Compare your answer:

The left eye can be given by $y = - (x + 2)^{2} + 50$ because the vertex is located at $(- 2, 50)$ and it opens downward.

Write an equation to represent the right eye in the smiley face.

Compare your answer:

The right eye can be given by $y = - (x - 2)^{2} + 50$ because the vertex is located at $(2, 50)$ and it opens downward.

What is the domain for the mouth? Write the domain as an inequality.

Compare your answer:

For the mouth, the domain is $- 4 \leq x \leq 4$ .

What is the domain for the left eye? Write the domain as an inequality.

Compare your answer:

For the left eye, the domain is $- 3 \leq x \leq - 1$ .

What is the domain for the right eye? Write the domain as an inequality.

Compare your answer:

For the right eye, the domain is $1 \leq x \leq 3$ .

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
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- Section URL: https://openstax.org/books/algebra-1/pages/7-17-4-transforming-the-parent-function

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