Main content

## SAT

### Course: SAT > Unit 6

Lesson 4: Passport to Advanced Math: lessons by skill- Solving quadratic equations | Lesson
- Interpreting nonlinear expressions | Lesson
- Quadratic and exponential word problems | Lesson
- Manipulating quadratic and exponential expressions | Lesson
- Radicals and rational exponents | Lesson
- Radical and rational equations | Lesson
- Operations with rational expressions | Lesson
- Operations with polynomials | Lesson
- Polynomial factors and graphs | Lesson
- Graphing quadratic functions | Lesson
- Graphing exponential functions | Lesson
- Linear and quadratic systems | Lesson
- Structure in expressions | Lesson
- Isolating quantities | Lesson
- Function Notation | Lesson

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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

### 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:

- Evaluate the function at several different values of x.
- Plot the input-output pairs as points in the x, y-plane.
- 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!

## How do I identify features of parabolas from quadratic functions?

### Forms & features of quadratic functions

### 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:

- Remember which equation form displays the relevant features as constants or coefficients.
- Rewrite the equation in a more helpful form if necessary.
- 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:

- Determine the features of the parabola.
- Identify the features shown in quadratic equation(s).
- Select a quadratic equation with the same features as the parabola.
- 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!

## How do I transform graphs of quadratic functions?

### Intro to parabola transformations

### 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!

## Your turn!

## 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?

- Is there going to be more lessons like these or is this the end, because so far it has been very helpful(30 votes)
- 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?(9 votes)
- yes, it is possible, you will need to use -b/2a for the x coordinate of the vertex and another formula k=c- b^2/4a for the y coordinate of the vertex. good luck, hope this helped(5 votes)

- 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)
- Also, remember not to stress out over it. Make sure to get a full nights. Good luck on your exam!(5 votes)

- Please help me get access to questions from "Graphing Quadratic Functions"(3 votes)
- how would i graph this though f(x)=2(x-3)^2-2(2 votes)
- 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)

- 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.(1 vote)
- 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.(2 votes)