# Graphs of Quadratic Functions | College Algebra (2022)

### Learning Outcomes

• Recognize characteristics of parabolas.
• Understand how the graph of a parabola is related to its quadratic function.

Curved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)

## Characteristics of Parabolas

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.

The $y$-intercept is the point at which the parabola crosses the $y$-axis. The $x$-intercepts are the points at which the parabola crosses the $x$-axis. If they exist, the $x$-intercepts represent the zeros, or roots, of the quadratic function, the values of $x$at which $y=0$.

### Example: Identifying the Characteristics of a Parabola

Determine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown below.

Show Solution

The general form of a quadratic function presents the function in the form

$f\left(x\right)=a{x}^{2}+bx+c$

where $a$, $b$, and $c$ are real numbers and $a\ne 0$. If $a>0$, the parabola opens upward. If $a<0$, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.

(Video) Graphing Quadratic Functions in Vertex & Standard Form - Axis of Symmetry - Word Problems

The axis of symmetry is defined by $x=-\dfrac{b}{2a}$. If we use the quadratic formula, $x=\dfrac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$, to solve $a{x}^{2}+bx+c=0$ for the $x$-intercepts, or zeros, we find the value of $x$halfway between them is always $x=-\dfrac{b}{2a}$, the equation for the axis of symmetry.

The figure below showsthe graph of the quadratic function written in general form as $y={x}^{2}+4x+3$. In this form, $a=1,\text{ }b=4$, and $c=3$. Because $a>0$, the parabola opens upward. The axis of symmetry is $x=-\dfrac{4}{2\left(1\right)}=-2$. This also makes sense because we can see from the graph that the vertical line $x=-2$ divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, $\left(-2,-1\right)$. The $x$-intercepts, those points where the parabola crosses the $x$-axis, occur at $\left(-3,0\right)$ and $\left(-1,0\right)$.

The standard form of a quadratic function presents the function in the form

$f\left(x\right)=a{\left(x-h\right)}^{2}+k$

where $\left(h,\text{ }k\right)$ is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.

## Given a quadratic function in general form, find the vertex of the parabola.

One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, $k$, and where it occurs, $h$. If we are given the general form of a quadratic function:

$f(x)=ax^2+bx+c$

We can define the vertex, $(h,k)$, by doing the following:

• Identify $a$, $b$, and $c$.
• Find $h$, the $x$-coordinate of the vertex, by substituting $a$ and $b$into $h=-\dfrac{b}{2a}$.
• Find $k$, the $y$-coordinate of the vertex, by evaluating $k=f\left(h\right)=f\left(-\dfrac{b}{2a}\right)$

### Example: Finding the Vertex of a Quadratic Function

Find the vertex of the quadratic function $f\left(x\right)=2{x}^{2}-6x+7$. Rewrite the quadratic in standard form (vertex form).

### Try It

Given the equation $g\left(x\right)=13+{x}^{2}-6x$, write the equation in general form and then in standard form.

Show Solution

(Video) Graphing Quadratic Functions (Precalculus - College Algebra 24)

## Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all $y$-values greater than or equal to the $y$-coordinate of the vertex or less than or equal to the $y$-coordinate at the turning point, depending on whether the parabola opens up or down.

### A General Note: Domain and Range of a Quadratic Function

The domain of any quadratic function is all real numbers.

The range of a quadratic function written in general form $f\left(x\right)=a{x}^{2}+bx+c$ with a positive $a$ value is $f\left(x\right)\ge f\left(-\frac{b}{2a}\right)$, or $\left[f\left(-\frac{b}{2a}\right),\infty \right)$; the range of a quadratic function written in general form with a negative $a$value is $f\left(x\right)\le f\left(-\frac{b}{2a}\right)$, or $\left(-\infty ,f\left(-\frac{b}{2a}\right)\right]$.

The range of a quadratic function written in standard form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ with a positive $a$value is $f\left(x\right)\ge k$; the range of a quadratic function written in standard form with a negative $a$value is $f\left(x\right)\le k$.

### How To: Given a quadratic function, find the domain and range.

1. The domain of any quadratic function as all real numbers.
2. Determine whether $a$ is positive or negative. If $a$is positive, the parabola has a minimum. If $a$is negative, the parabola has a maximum.
3. Determine the maximum or minimum value of the parabola, $k$.
4. If the parabola has a minimum, the range is given by $f\left(x\right)\ge k$, or $\left[k,\infty \right)$. If the parabola has a maximum, the range is given by $f\left(x\right)\le k$, or $\left(-\infty ,k\right]$.

### Example: Finding the Domain and Range of a Quadratic Function

Find the domain and range of $f\left(x\right)=-5{x}^{2}+9x - 1$.

Show Solution

### Try It

Find the domain and range of $f\left(x\right)=2{\left(x-\dfrac{4}{7}\right)}^{2}+\dfrac{8}{11}$.

Show Solution

The standard form of a quadratic function presents the function in the form

$f\left(x\right)=a{\left(x-h\right)}^{2}+k$

where $\left(h,k\right)$ is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.

(Video) College Algebra 3.1 Graphing quadratic functions

The standard form is useful for determining how the graph is transformed from the graph of $y={x}^{2}$. The figure belowis the graph of this basic function.

## Shift Up and Down by Changing the Value of $k$

You can represent a vertical (up, down) shift of the graph of $f(x)=x^2$ by adding or subtracting a constant, $k$.

$f(x)=x^2 + k$

If $k>0$, the graph shifts upward, whereas if $k<0$, the graph shifts downward.

### Example

Determine the equation for the graph of$f(x)=x^2$ that has been shifted up 4 units. Also, determine the equation for the graph of$f(x)=x^2$ that has been shifted down 4 units.

Show Solution

## Shift left and right by changing the value of $h$

You can represent a horizontal (left, right) shift of the graph of $f(x)=x^2$ by adding or subtracting a constant, $h$, to the variable $x$, before squaring.

$f(x)=(x-h)^2$

If $h>0$, the graph shifts toward the right and if $h<0$, the graph shifts to the left.

### Example

Determine the equation for the graph of$f(x)=x^2$ that has been shifted right 2 units. Also, determine the equation for the graph of$f(x)=x^2$ that has been shifted left 2 units.

Show Solution

## Stretch or compress by changing the value of $a$.

You can represent a stretch or compression (narrowing, widening)of the graph of $f(x)=x^2$ bymultiplying the squared variable by a constant, $a$.

(Video) College Algebra Section 4.3 Quadratic Functions

$f(x)=ax^2$

The magnitude of $a$indicates the stretch of the graph. If $|a|>1$, the point associated with a particular $x$-value shifts farther from the $x$axis, so the graph appears to become narrower, and there is a vertical stretch. But if $|a|<1$, the point associated with a particular $x$-value shifts closer to the $x$axis, so the graph appears to become wider, but in fact there is a vertical compression.

### Example

Determine the equation for the graph of$f(x)=x^2$ that has been compressed vertically by a factor of $\frac{1}{2}$. Also, determine the equation for the graph of$f(x)=x^2$ that has been vertically stretched by a factor of 3.

Show Solution

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.

\begin{align}&a{\left(x-h\right)}^{2}+k=a{x}^{2}+bx+c\\ &a{x}^{2}-2ahx+\left(a{h}^{2}+k\right)=a{x}^{2}+bx+c \end{align}

For the two sides to be equal, the corresponding coefficients must be equal. In particular, the coefficients of $x$ must be equal.

$-2ah=b,\text{ so }h=-\dfrac{b}{2a}$.

This is the $x$ coordinate of the vertexr and $x=-\dfrac{b}{2a}$ is theaxis of symmetry we defined earlier. Setting the constant terms equal gives us:

\begin{align}a{h}^{2}+k&=c \\[2mm] k&=c-a{h}^{2} \\ &=c-a-{\left(\dfrac{b}{2a}\right)}^{2} \\ &=c-\dfrac{{b}^{2}}{4a} \end{align}

In practice, though, it is usually easier to remember that $h$is the output value of the function when the input is $h$, so $f\left(h\right)=f\left(-\dfrac{b}{2a}\right)=k$.

### Try It

A coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?

(Video) Learn how to graph a quadratic

(credit: modification of work by Dan Meyer)

## Key Concepts

• A polynomial function of degree two is called a quadratic function.
• The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
• The axis of symmetry is the vertical line passing through the vertex.
• Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
• The vertex can be found from an equation representing a quadratic function.
• The domain of a quadratic function is all real numbers. The range varies with the function.

## Glossary

axis of symmetry
a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by $x=-\dfrac{b}{2a}$.
general form of a quadratic function
the function that describes a parabola, written in the form $f\left(x\right)=a{x}^{2}+bx+c$, where $a$, $b$, and $c$are real numbers and $a\ne 0$.
standard form of a quadratic function
the function that describes a parabola, written in the form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$, where $\left(h,\text{ }k\right)$ is the vertex.

## FAQs

### What graph represents a quadratic function? ›

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function.

### What are the types of quadratic graphs? ›

There are three commonly-used forms of quadratics:
• Standard Form: y = a x 2 + b x + c y=ax^2+bx+c y=ax2+bx+c.
• Factored Form: y = a ( x − r 1 ) ( x − r 2 ) y=a(x-r_1)(x-r_2) y=a(x−r1)(x−r2)
• Vertex Form: y = a ( x − h ) 2 + k y=a(x-h)^2+k y=a(x−h)2+k.
Mar 1, 2022

### What are examples of quadratic functions? ›

The quadratic function equation is f(x) = ax2 + bx + c, where a ≠ 0. Let us see a few examples of quadratic functions: f(x) = 2x2 + 4x - 5; Here a = 2, b = 4, c = -5. f(x) = 3x2 - 9; Here a = 3, b = 0, c = -9.

### How do you know if a graph is a quadratic function? ›

Graphs. A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

### Is the graph of a quadratic function always a parabola? ›

A quadratic function is a function of degree two. The graph of a quadratic function is a parabola.

### Which two types of graphs are functions? ›

• Linear graphs are produced by linear functions of this form:
• Power graphs are produced by functions with only one term and a power. ...
• Quadratics are functions where the highest power is two.
• Polynomials are a more general function than a quadratic and allow for higher powers that are still whole numbers.
Sep 16, 2021

### What are 5 methods of solving a quadratic equation? ›

There are several methods you can use to solve a quadratic equation: Factoring Completing the Square Quadratic Formula Graphing
• Factoring.
• Completing the Square.
• Graphing.

### How do you graph an equation step by step? ›

Steps for graphing an equation using the slope and y-intercept:
1. Find the y-intercept = b of the equation y = mx + b.
2. Plot the y-intercept. The point will be (0, b).
3. Find the slope=m of the equation y = mx + b.
4. Make a single step, using the rise and run from the slope. ...
5. Connect those two points with your line.

### How do you graph a quadratic equation without a table? ›

graphing a parabola without a table - YouTube

### How do you solve a quadratic function? ›

1. Put all terms on one side of the equal sign, leaving zero on the other side.
2. Factor.
3. Set each factor equal to zero.
4. Solve each of these equations.

### What's the difference between parabola and quadratic? ›

A parabola is a curve with a line of symmetry at the maximum or minimum. Quadratic graphs always follow the equation ax^2 + bx + c = 0, where "a" cannot equal 0. If "a" is greater than 0, then the parabola opens upward and we can measure a minimum.

### What does a quadratic graph look like? ›

The graph of a quadratic function is called a parabola and has a curved shape. One of the main points of a parabola is its vertex. It is the highest or the lowest point on its graph. You can think of like an endpoint of a parabola.

### Why do we need to study quadratic equation? ›

Answer: In daily life we use quadratic formula as for calculating areas, determining a product's profit or formulating the speed of an object. In addition, quadratic equations refer to an equation that has at least one squared variable.

### What is a parabola graph? ›

A parabola graph depicts a U-shaped curve drawn for a quadratic function. In Mathematics, a parabola is one of the conic sections, which is formed by the intersection of a right circular cone by a plane surface. It is a symmetrical plane U-shaped curve.

### What is the fastest way to solve a quadratic equation? ›

How to Solve Quadratic Equations in SECONDS - Quick & Easy Trick

### What equation is not quadratic? ›

In general, a quadratic equation: must contain an x2 term. must NOT contain terms with degrees higher than x2 eg. x3, x4 etc.

### How do you graph a function? ›

Graphing a Basic Function - YouTube

### How do you find the quadratic function? ›

The quadratic function f(x) = a(x - h)2 + k, a not equal to zero, is said to be in standard form. If a is positive, the graph opens upward, and if a is negative, then it opens downward. The line of symmetry is the vertical line x = h, and the vertex is the point (h,k).

### What are the basic graphs? ›

A basic two-dimensional graph consists of a vertical and a horizontal line that intersects at a point called origin. The horizontal line is the x axis, the vertical line is the y axis. In simple line graphs, the x and y axes are each divided into evenly spaced subdivisions that are assigned to numerical values.

### How do you plot a graph? ›

How to Plot a Graph on Paper in Science - YouTube

### What does a quadratic graph look like? ›

The graph of a quadratic function is called a parabola and has a curved shape. One of the main points of a parabola is its vertex. It is the highest or the lowest point on its graph. You can think of like an endpoint of a parabola.

### Do all quadratic graphs have roots? ›

A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one, two, or zero roots.

### How do you graph a parabola? ›

How to Graph Parabolas - YouTube

### What is a function graph? ›

The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation.

### How do you graph functions on a calculator? ›

Graphing Calculator - Enter and Graph Functions - YouTube

### How do you graph functions on a graphing calculator? ›

The first step in any graphing problem is to draw the graph. On the TI-83 and TI-84, this is done by going to the function screen by pressing the “Y=” button and entering the function into one of the lines. After the function has been entered, press the “GRAPH” button, and the calculator will draw the graph for you.

A quadratic function is a polynomial function of degree 2 which can be written in the general form,

Find the vertex of a parabola by completing the square.. For any parabola, we will find the vertex and y -intercept.. One way to do this is to first use x=−b2a to find the x -value of the vertex and then substitute this value in the function to find the corresponding y -value.. Next, find the vertex.. To find it, first find the x -value of the vertex.. Rewrite in vertex form and determine the vertex: f(x)=x2+4x+9.. Rewrite in vertex form and determine the vertex: f(x)=2x2−4x+8.. Rewrite in vertex form and determine the vertex: f(x)=−2x2−12x+3.. When graphing a parabola always find the vertex and the y -intercept.. An alternate approach to finding the vertex is to rewrite the quadratic function in the form f ( x ) = a ( x − h ) 2 + k . When in this form, the vertex is ( h , k ) and can be read directly from the equation.. Find the vertex and the y -intercept.. Rewrite in vertex form y=a(x−h)2+k and determine the vertex.. Find the vertex and the y -intercept.

If you’re taking the CLEP College Algebra test in a few weeks or months, you might be anxious about how to remember ALL the different formulas and math concepts and recall them during the test. The CLEP College Algebra covers a wide range of

Following is a quick formula reference sheet that lists all important CLEP College Algebra every formula in this CLEP College Algebra Formula Cheat Sheet , you will save yourself valuable time on the test and probably get a few extra questions correct.. A number expressed in the form $$\frac{a}{b}$$Adding and Subtracting with the same denominator:$$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$$$$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$$Adding and Subtracting with the different denominator:$$\frac{a}{b}+\frac{c}{d}=\frac{ad+cb}{bd}$$$$\frac{a}{b}-\frac{c}{d}=\frac{ad-cb}{bd}$$Multiplying and Dividing Fractions:$$\frac{a}{b} × \frac{c}{d}=\frac{a×c}{b×d}$$$$\frac{a}{b} ÷ \frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}$$. A number composed of a whole number and fraction.. Factor a number means to break it up into numbers that can be multiplied together to get the original number.. $$\{…,-3,-2,-1,0,1,2,3,…\}$$Includes: zero, counting numbers, and the negative of the counting numbers. All numbers that are on number line.. Refers to the number of times a number is multiplied by itself.$$8 = 2 × 2 × 2 = 2^3$$. It is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.In scientific notation all numbers are written in this form: $$m \times 10^n$$ Scientific notation :$$5×10^0$$$$-2.5×10^4$$$$5×10^{-1}$$$$2,122456×10^3$$

The term quadratic comes from the word quadrate meaning square or rectangular. Similarly, one of the definitions of the term quadratic is a square. In an algebraic sense, the definition of something quadratic involves the square and no higher power of an unknown quantity; second degree. So, for our purposes, we will be working with quadratic equations which mean that the highest degree we'll be encountering is a square. Normally, we see the standard quadratic equation written as the sum of three terms set equal to zero. Simply, the three terms include one that has an x2, one has an x, and one term is "by itself" with no x2 or x.

Let's graph the equation again.. Graph of y = x 2. Let's trying graphing another parabola where a = 1, b = -2. and c = 0.. Now let's try graphing the parabola: y = -3x 2 +. x + 1.. The vertex for this parabola would be (3, -11).. Graph of y = 2x 2 - 12x + 7. What about the graph of the equation, y = 2x 2 ?. Graph of the parabolas, p(x) = (x - 4) 2 and q(x) = (x - 4) 2 + 7. Example : The vertex of the parabola y = 3(x - 1) 2 + 8 is (1, 8).. In the first equation, y = 4(x + 3) 2 + 4, the vertex. is (h, k) or (-3, 4). Using our same equations, y = 4x 2 + 24x + 40 and. y = 4(x + 3) 2 + 4, we already know that the vertex. is (-3, 4) in both of them.

## Videos

1. Graphing a parabola using roots and vertex | Quadratic equations | Algebra I | Khan Academy
(ThinkwellVids)
(StudyPug)
4. Algebra 1 - Intro to Understanding and Graphing Quadratic Functions
(iteachalgebra)
5. Graphs of Quadratic Functions: An Application (Algebra I)
(CK-12 Foundation)
6. Graph Quadratic Equations without a Calculator - Step-By-Step Approach
(PreMath)

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