Symmetry in Graphs of Quadratic Functions

New York State Common Core Math Algebra I, Module 4, Lesson 8

Worksheets for Algebra 1

Videos, examples, and solutions to help Algebra I students learn how to examine quadratic equations in two variables represented graphically on a coordinate plane and recognize the symmetry of the graph. They explore key features of graphs of quadratic functions: y-intercept and x-intercepts, the vertex, the axis of symmetry, increasing and decreasing intervals, negative and positive intervals, and end behavior. They sketch graphs of quadratic functions as a symmetric curve with a highest or lowest point corresponding to its vertex and an axis of symmetry passing through the vertex.

Graph Vocabulary

Axis of Symmetry: Given a quadratic function in standard form, f(x) = ax² + bx + c, the vertical line given by the graph of the equation, x = -b/(2a) , is called the axis of symmetry of the graph of the quadratic function.

Vertex: The point where the graph of a quadratic function and its axis of symmetry intersect is called the vertex.

End Behavior of a Graph: Given a quadratic function in the form f(x) = ax² + bx + c (or f(x) = a(x-h)² + k ), the quadratic function is said to open up if a > 0 and open down if a < 0.

If a > 0, then f has a minimum at the x-coordinate of the vertex, i.e. f, is decreasing for x-values less than (or to the left of) the vertex, and f is increasing for x values greater than (or to the right of) the vertex.
If a > 0, then f has a maximum at x-coordinate of the vertex, i.e. f, is increasing for x-values less than (or to the left of) the vertex, and f is decreasing for x-values greater than (or to the right of) the vertex.

Exploratory Challenge 3

Below you see only one side of the graph of a quadratic function. Complete the graph by plotting three additional points of the quadratic function. Explain how you found these points then fill in the table on the right.

a. What are the coordinates of the -intercepts?
b. What are the coordinates of the -intercept?
c. What are the coordinates of the vertex? Is it a minimum or a maximum?
d. If we knew the equation for this curve, what would the sign of the leading coefficient be?
e. Verify that the average rate of change for interval, -3 ≤ x ≤ -2, [-3, -2] is 5. Show your steps.
f. What is the axis of symmetry?

Lesson 8 Summary

Quadratic functions create a symmetrical curve with its highest (maximum) or lowest (minimum) point corresponding to its vertex and an axis of symmetry passing through it when graphed. The x-coordinate of the vertex is the average of the x-coordinates of the zeros or any two symmetric points on the graph.

When the leading coefficient is a negative number, the graph opens down and its end behavior is that both ends move towards negative infinity. If the leading coefficient is positive, the graph opens up and both ends move towards positive infinity.

Lesson 8 Problem Set Sample Solutions

3. Consider the following key features discussed in this lesson for the four graphs of quadratic functions below: x-intercepts, y-intercept, line of symmetry, vertex, and end behavior.

a. Which key features of a quadratic function do graphs and have in common? Which features are not shared?
b. Compare the graphs and and explain the differences and similarities between their key features.
c. Compare the graphs and and explain the differences and similarities between their key features.
d. What do all four of the graphs have in common?

4. Use the symmetric properties of quadratic functions to sketch the graph of the function below, given these points and given that the vertex of the graph is the point (0,5).

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