In this resource, you will analyze the graphs of quadratic functions. In the first section, you will focus on the direction in which a parabola opens and the width of the parabola itself.

This is exciting! A smile is forming on your face.

The smile on your face could be considered a possible graph of a quadratic function. Click on the graph below.

• In which direction do both parabolas in the graph open?

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  Both parabolas open upward.

• What does the direction in which the parabolas tell you about the sign of the value of a in the function y = ax² that would generate each parabola?

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  Because both parabolas open upward, the sign of the value of a in the function generating each parabola is positive.

• How does the parabola representing the smile compare to the original parabola, which is the quadratic parent function, y = x²?

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  The parabola representing the smile is more narrow than the parent function.

• How does the value of a in the function y = ax² for the parabola representing the smile compare to the value of a for the parent function, y = x²?

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  The value of a for the parabola representing the smile is greater than 1 and is greater than the value of a for the parent function.

• Is the vertex for each parabola a minimum value or a maximum value? How can you tell?

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  The vertex for each parabola is a minimum value because the y-value for the vertex is less than the y-values for the remaining points on each parabola.

Place the four smiles in order from the least value of a to the greatest value of a. Drag and drop each smile to its appropriate place.

View the animation below to see parabolas in a famous marketing symbol. Click on the words "famous landmark" to start the animation, and then use the animation to answer the questions that follow.

• In what direction do the parabolas in the "Golden Arches" symbol appear to open?

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  The parabolas open downward.

• If you were to write an equation representing each parabola in the "Golden Arches," would the value of a be positive or negative? How do you know?

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  The value of a would be negative because the parabolas open downward.

• Are the vertices for each parabola in the "Golden Arches" maximum values or minimum values? How do you know?

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  The vertices are maximum values because all other y-values in each parabola would be less than the vertex at the top of the arch.

Practice

The graphs below are all of the form y = ax².

1. Which of the graphs have a value of a < 0?

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   If the value of a is less than 0, will the graph open upward or downward?

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   Graphs I and IV

2. Which graph appears to have the largest |a|?

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   Which graph appears to be the narrowest?

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   Graph II

3. List the graphs in order from the greatest value of a to the least value of a.

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   The narrowest graph that opens upward will have the greatest value of a, and the narrowest graph that opens downward will have the least value of a.

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   Graph II, Graph III, Graph I, Graph IV