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Quadratic graphs | Lesson

A guide to quadratic graphs on the digital SAT

What are quadratic functions?

In a quadratic function, the
output
of the function is based on an expression in which the
input to the second power
is the highest power term. For example, f, left parenthesis, x, right parenthesis, equals, x, squared, plus, 2, x, plus, 1 is a quadratic function, because in the highest power term, the x is raised to the second power.
Unlike the graphs of linear functions, the graphs of quadratic functions are nonlinear: they don't look like straight lines. Specifically, the graphs of quadratic functions are called parabolas.
The graph of y=x^2+2x+1 is a parabola.
In this lesson, we'll learn to:
  1. Graph quadratic functions
  2. Identify the features of quadratic functions
  3. Rewrite quadratic functions to showcase specific features of graphs
  4. Transform quadratic functions
You can learn anything. Let's do this!

How do I graph parabolas, and what are their features?

Parabolas intro

Khan Academy video wrapper
Parabolas introSee video transcript

What are the features of a parabola?

All parabolas have a y-intercept, a
vertex
, and open either upward or downward.
Two parabolas are graphed in the xy-plane. The parabola represented by the equation y=(x+2)^2+4 opens upward, has a vertex located at (-2, 4), and a y-intercept located at (0, 8). The parabola represented by the equation y=-0.5x^2+2x+3 opens downward, has a vertex located at (2, 5), and a y-intercept located at (0, 3).
Since the vertex is the point at which a parabola changes from increasing to decreasing or vice versa, it is also either the maximum or minimum y-value of the parabola.
  • If the parabola opens upward, then the vertex is the lowest point on the parabola.
  • If the parabola opens downward, then the vertex is the highest point on the parabola.
A parabola can also have zero, one, or two x-intercepts.
Note: the terms "zero" and "root" are used interchangeably with "x-intercept". They all mean the same thing!
Three parabolas are graphed in the xy-plane, each with a different number of x-intercepts. The parabola represented by the equation y=-2(x+3)(x+7) opens downward and has two x-intercepts, (-7, 0) and (-3, 0). The parabola represented by the equation y=x^2 opens upward and has one x-intercept, (0, 0). The parabola represented by the equation y=-0.25(x-5)^2-3 opens downward and has no x-intercept.
Parabolas also have vertical symmetry along a vertical line that passes through the vertex.
For example, if a parabola has a vertex at left parenthesis, 2, comma, 0, right parenthesis, then the parabola has the same y-values at x, equals, 1 and x, equals, 3, at x, equals, 0 and x, equals, 4, and so on.
A parabola is graphed in the xy-plane. The parabola opens upward and has a vertex at (2,0). A vertical line represented by x=2, is drawn through the vertex, establishing vertical symmetry. The parabola contains the symmetrical points (1,3) and (3,3), as well as (0, 4) and (4, 4).
To graph a quadratic function:
  1. Evaluate the function at several different values of x.
  2. Plot the input-output pairs as points in the x, y-plane.
  3. Sketch a parabola that passes through the points.

Example: Graph f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 3 in the x, y-plane.

Try it!

TRY: match the features of the parabola to their coordinates
The graph of y=-x^2+4x+5 is a parabola that opens downward. The graph crosses the y-axis at (0, 5), changes directions from increasing to decreasing at (2, 9), and crosses the x-axis at (-1, 0) and (5, 0).
The graph of y, equals, minus, x, squared, plus, 4, x, plus, 5 is shown above. Match the features of the graph with their coordinates.


How do I identify features of parabolas from quadratic functions?

Forms & features of quadratic functions

Khan Academy video wrapper
Forms & features of quadratic functionsSee video transcript

Standard form, factored form, and vertex form: What forms do quadratic equations take?

For all three forms of quadratic equations, the coefficient of the x, squared-term, start color #7854ab, a, end color #7854ab, tells us whether the parabola opens upward or downward:
  • If start color #7854ab, a, end color #7854ab, is greater than, 0, then the parabola opens upward.
  • If start color #7854ab, a, end color #7854ab, is less than, 0, then the parabola opens downward.
The magnitude of start color #7854ab, a, end color #7854ab also describes how steep or shallow the parabola is. Parabolas with larger magnitudes of start color #7854ab, a, end color #7854ab are more steep and narrow compared to parabolas with smaller magnitudes of start color #7854ab, a, end color #7854ab, which tend to be more shallow and wide.
The graph below shows the graphs of y, equals, start color #7854ab, a, end color #7854ab, x, squared for various values of start color #7854ab, a, end color #7854ab.
Four parabolas are graphed in the xy-plane, each represented by the equation y=ax^2 for a different value of a. The four parabolas share a vertex at (0,0) but have different widths. The parabola corresponding to a=0.1 is the most shallow and wide, and the parabolas corresponding to a=0.5, a=1, and a=2 are increasingly steep and narrow.
The standard form of a quadratic equation, y, equals, start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170, shows the y-intercept of the parabola:
  • The y-intercept of the parabola is located at left parenthesis, 0, comma, start color #208170, c, end color #208170, right parenthesis.
The factored form of a quadratic equation, y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, shows the x-intercept(s) of the parabola:
  • x, equals, start color #ca337c, b, end color #ca337c and x, equals, start color #208170, c, end color #208170 are solutions to the equation start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis, equals, 0.
  • The x-intercepts of the parabola are located at left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis and left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis.
  • The terms x-intercept, zero, and root can be used interchangeably.
The vertex form of a quadratic equation, y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, h, end color #ca337c, right parenthesis, squared, plus, start color #208170, k, end color #208170, reveals the vertex of the parabola.
  • The vertex of the parabola is located at left parenthesis, start color #ca337c, h, end color #ca337c, comma, start color #208170, k, end color #208170, right parenthesis.
To identify the features of a parabola from a quadratic equation:
  1. Remember which equation form displays the relevant features as constants or coefficients.
  2. Rewrite the equation in a more helpful form if necessary.
  3. Identify the constants or coefficients that correspond to the features of interest.

Example: What are the zeros of the graph of f, left parenthesis, x, right parenthesis, equals, x, squared, plus, 7, x, plus, 12 ?

To match a parabola with its quadratic equation:
  1. Determine the features of the parabola.
  2. Identify the features shown in quadratic equation(s).
  3. Select a quadratic equation with the same features as the parabola.
  4. Plug in a point that is not a feature from Step 2 to calculate the coefficient of the x, squared-term if necessary.

Example:
A parabola in the xy-plane opens downward, has a vertex located at (-3, 3), and has zeros located at (-4, 0) and (-2, 0).
What is a possible equation for the parabola shown above?

Try it!

TRY: determine the feature of a parabola from its equation
The graph of the equation y, equals, start fraction, 1, divided by, 2, end fraction, x, squared, minus, 7 is a parabola that opens
because the coefficient of x, squared is
.


TRY: determine the feature of a parabola from its equation
The
of the graph of f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 2, right parenthesis, squared, plus, 7 is located at the point left parenthesis, minus, 2, comma, 7, right parenthesis. 7 is the
y-value of the graph.


TRY: determine the feature of a parabola from its equation
y, equals, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, minus, 4, right parenthesis is the
form of a quadratic equation. The
of the graph can be identified as constants in the equation.


How do I rewrite quadratic functions to reveal specific features of parabolas?

Equivalent forms of quadratic functions

When we're given a quadratic function, we can rewrite the function according to the features we want to display:
  • y-intercept: standard form
  • x-intercept(s): factored form
  • Vertex: vertex form
When rewriting a quadratic function to display specific graphical features:
  1. Choose the appropriate quadratic form based on the graphic feature to be displayed.
  2. Rewrite the given quadratic expression as an equivalent expression in the form identified in Step 1.

Example:
A parabola is graphed in the xy-plane. The parabola opens upward and contains the points (-1, 0), (0, -5), (2, -9), and (5, 0).
The graph of y, equals, x, squared, minus, 4, x, minus, 5 is shown above. Write an equivalent equation from which the coordinates of the vertex can be identified as constants in the equation.

Try it!

Try: identify the appropriate forms to use
A parabola is graphed in the xy-plane. The parabola opens upward and contains the points (-1, 0), (0, -3), (1,-4), and (3, 0).
A graph of the quadratic equation y, equals, x, squared, minus, 2, x, minus, 3 is shown in the x, y-plane above.
The vertex of the parabola is located at
. To show the coordinates of the vertex as constants or coefficients, we should use the
form of the quadratic equation, y, equals, left parenthesis, x, minus, 1, right parenthesis, squared, minus, 4.
The x-intercepts of the parabola are
. To show the x-intercepts as constants or coefficients, we should use the
form of the quadratic equation, y, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis.


How do I transform graphs of quadratic functions?

Intro to parabola transformations

Khan Academy video wrapper
Intro to parabola transformationsSee video transcript

Translating, stretching, and reflecting: How does changing the function transform the parabola?

We can use function notation to represent the translation of a graph in the x, y-plane. If the graph of y, equals, f, left parenthesis, x, right parenthesis is graphed in the x, y-plane and c is a positive constant:
  • The graph of y, equals, f, left parenthesis, x, minus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the right by c units.
  • The graph of y, equals, f, left parenthesis, x, plus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the left by c units.
  • The graph of y, equals, f, left parenthesis, x, right parenthesis, plus, c is the graph of f, left parenthesis, x, right parenthesis shifted up by c units.
  • The graph of y, equals, f, left parenthesis, x, right parenthesis, minus, c is the graph of f, left parenthesis, x, right parenthesis shifted down by c units.
The graph below shows the graph of the quadratic function f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 3 alongside various translations:
  • The graph of start color #7854ab, f, left parenthesis, x, minus, 4, right parenthesis, equals, left parenthesis, x, minus, 4, right parenthesis, squared, minus, 3, end color #7854ab translates the graph of 4 units to the right.
  • The graph of start color #ca337c, f, left parenthesis, x, plus, 6, right parenthesis, equals, left parenthesis, x, plus, 6, right parenthesis, squared, minus, 3, end color #ca337c translates the graph 6 units to the left.
  • The graph of start color #208170, f, left parenthesis, x, right parenthesis, plus, 5, equals, x, squared, plus, 2, end color #208170 translates the graph 5 units up.
  • The graph of start color #a75a05, f, left parenthesis, x, right parenthesis, minus, 3, equals, x, squared, minus, 6, end color #a75a05 translates the graph 3 units down.
Five parabolas are graphed in the xy-plane. One of the parabolas represents the equation y=x^2-3, and the other four parabolas represent the graph of y=x^2-3 translated right, left, up, and down.
We can also represent stretching and reflecting graphs algebraically. If the graph of y, equals, f, left parenthesis, x, right parenthesis is graphed in the x, y-plane and c is a positive constant:
  • The graph of y, equals, minus, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the x-axis.
  • The graph of y, equals, f, left parenthesis, minus, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the y-axis.
  • The graph of y, equals, c, dot, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis stretched vertically by a factor of c.
The graph below shows the graph of the quadratic function f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 2, x, minus, 2 alongside various transformations:
  • The graph of start color #7854ab, minus, f, left parenthesis, x, right parenthesis, equals, minus, x, squared, plus, 2, x, plus, 2, end color #7854ab is the graph of f, left parenthesis, x, right parenthesis reflected across the x-axis.
  • The graph of start color #ca337c, f, left parenthesis, minus, x, right parenthesis, equals, x, squared, plus, 2, x, minus, 2, end color #ca337c is the graph of f, left parenthesis, x, right parenthesis reflected across the y-axis.
  • The graph of start color #208170, 3, dot, f, left parenthesis, x, right parenthesis, equals, 3, x, squared, minus, 6, x, minus, 6, end color #208170 is the graph of f, left parenthesis, x, right parenthesis stretched vertically by a factor of 3.
Four parabolas are graphed in the xy-plane. One of the parabolas represents the equation y=x^2-2x-2, and the other three parabolas represent the graph of y=x^2-2x-2 reflected across the x-axis, reflected across the y-axis, and stretched vertically by a factor of 3.

Try it!

TRY: Shift a parabola
Compared to the graph of y, equals, x, squared, the graph of y, equals, left parenthesis, x, plus, 4, right parenthesis, squared, plus, 3 is shifted 4 units
and 3 units
.


Try: reflect a parabola
The graph of y, equals, f, left parenthesis, x, right parenthesis is a parabola that opens upward and has a vertex located at left parenthesis, 2, comma, 1, right parenthesis.
The graph of y, equals, f, left parenthesis, minus, x, right parenthesis has a vertex located at
.
The graph of y, equals, minus, f, left parenthesis, x, right parenthesis is a parabola that opens
.


Your turn!

Practice: identify the features of a parabola
If the function f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, squared, minus, 11 is graphed in the x, y-plane, what are the coordinates of the vertex?
Choose 1 answer:
Choose 1 answer:


Practice: match a parabola to a quadratic function
A parabola is graphed in the xy-plane. The parabola opens downward and contains the points (-1, 0), (1,4), and (3, 0).
Function f is graphed in the x, y-plane above. Which of the following could be f ?
Choose 1 answer:
Choose 1 answer:


Practice: identify the graphical feature displayed as a constant or coefficient
If y, equals, left parenthesis, x, minus, 2, right parenthesis, squared, plus, 6 is graphed in the x, y-plane. which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?
Choose 1 answer:
Choose 1 answer:

Things to remember

Forms of quadratic equations

Standard form: A parabola with the equation y, equals, start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170 has its y-intercept located at left parenthesis, 0, comma, start color #208170, c, end color #208170, right parenthesis.
Factored form: A parabola with the equation y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, x, minus, start color #208170, c, end color #208170, right parenthesis has its x-intercept(s) located at left parenthesis, start color #ca337c, b, end color #ca337c, comma, 0, right parenthesis and left parenthesis, start color #208170, c, end color #208170, comma, 0, right parenthesis.
Vertex form: A parabola with the equation y, equals, start color #7854ab, a, end color #7854ab, left parenthesis, x, minus, start color #ca337c, h, end color #ca337c, right parenthesis, squared, plus, start color #208170, k, end color #208170 has its vertex located at left parenthesis, start color #ca337c, h, end color #ca337c, comma, start color #208170, k, end color #208170, right parenthesis.
When we're given a quadratic function, we can rewrite the function according to the features we want to display:
  • y-intercept: standard form
  • x-intercept(s): factored form
  • Vertex: vertex form

Transformations

If the graph of y, equals, f, left parenthesis, x, right parenthesis is graphed in the x, y-plane and c is a positive constant:
  • The graph of y, equals, f, left parenthesis, x, minus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the right by c units.
  • The graph of y, equals, f, left parenthesis, x, plus, c, right parenthesis is the graph of f, left parenthesis, x, right parenthesis shifted to the left by c units.
  • The graph of y, equals, f, left parenthesis, x, right parenthesis, plus, c is the graph of f, left parenthesis, x, right parenthesis shifted up by c units.
  • The graph of y, equals, f, left parenthesis, x, right parenthesis, minus, c is the graph of f, left parenthesis, x, right parenthesis shifted down by c units.
  • The graph of y, equals, minus, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the x-axis.
  • The graph of y, equals, f, left parenthesis, minus, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis reflected across the y-axis.
  • The graph of y, equals, c, dot, f, left parenthesis, x, right parenthesis is the graph of f, left parenthesis, x, right parenthesis stretched vertically by a factor of c.

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