If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Graphing quadratic functions | Lesson

What are quadratic functions, and how frequently do they appear on the test?

In a quadratic function, the
of the function is based on an expression in which the
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.
On your official SAT, you'll likely see 2 to 4 questions that test your understanding of the connection between quadratic functions and parabolas.
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
, and open either upward or downward.
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!
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.
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, equals, minus, x, squared, plus, 4, x, plus, 5 is shown above. Match the features of the graph with their coordinates.
1


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.
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:
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 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.
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.

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:


Practice: match a parabola to a quadratic function
Function f is graphed in the x, y-plane above. Which of the following could be f ?
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.

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.

Want to join the conversation?

  • marcimus pink style avatar for user IZH1
    Is it possible to find the vertex of the parabola using the equation -b/2a as well as the other equations listed in the article?
    (10 votes)
    Default Khan Academy avatar avatar for user
  • female robot amelia style avatar for user namra guleen
    my sat is on 13 of march(probably after5 days ) n i'm craming over maths I just need 500 to 600 score for math so which topics should I focus on more ??
    (2 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Gracie Lynch
    How do you get the formula from looking at the parabola? I am having trouble when I try to work backward with what he said.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Hecretary Bird
      You can get the formula from looking at the graph of a parabola in two ways: Either by considering the roots of the parabola or the vertex. A parabola is not like a straight line that you can find the equation of if you have two points on the graph, because there are multiple different parabolas that can go through a given set of two points. Instead you need three points, or the vertex and a point.

      In the last practice problem on this article, you're asked to find the equation of a parabola. Think about how you can find the roots of a quadratic equation by factoring. The same principle applies here, just in reverse. You can figure out the roots (x-intercepts) from the graph, and just put them together as factors to make an equation. In this form, the equation for a parabola would look like y = a(x - m)(x - n). "a" is a coefficient (responsible for vertically stretching/flipping the parabola and thus doesn't affect the roots), and the roots of the graph are at x = m and x = n. Because the graph in the problem has roots at 3 and -1, our equation would look like y = a(x + 1)(x - 3). The only one that fits this is answer choice B), which has "a" be -1.

      You can also find the equation of a quadratic equation by finding the coordinates of the vertex from a graph, then plugging that into vertex form, and then picking a point on the parabola to use in order to solve for your "a" value. Thirdly, I guess you could also use three separate points to put in a system of three equations, which would let you solve for the "a", "b", and "c" in the standard form of a quadratic, but that's too much work for the SAT.
      (3 votes)
  • blobby green style avatar for user Aranea
    Please help me get access to questions from "Graphing Quadratic Functions"
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user 0070193150
    how would i graph this though f(x)=2(x-3)^2-2
    (2 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Hecretary Bird
      Following the steps in the article, you would graph this function by following the steps to transform the parent function of y = x^2.
      Here, we see that 3 is subtracted from x inside the parentheses, which means that we translate right by 3. We subtract 2 from the final answer, so we move down by 2. Our vertex will then be right 3 and down 2 from the normal vertex (0,0), at (3, -2).
      From here, we see that there's a coefficient outside the parentheses, which means we vertically stretch the function by a factor of 2.
      The easiest way to graph this would be to find the vertex and direction that it opens, and then plug in a point for x and see what you get for y. If we plugged in 5, we would get y = 4. You can put that point in the graph as well, and then draw a parabola that has that vertex and goes through the second point.
      (2 votes)