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# Quadratic Functions and Their Graphs

## Identify the intercepts, vertex, and axis of symmetry

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Quadratic Functions and Their Graphs

What if you had a quadratic function like 5+2x3x2\begin{align*}5 + 2x - 3x^2\end{align*}? What would its graph look like? Would the graph of 5+2xx2\begin{align*}5 + 2x - x^2\end{align*} be wider or narrower than it? After completing this Concept, you'll be able to graph and compare graphs of quadratic functions like these.

### Watch This

CK-12 Foundation: 1001S Graphs of Quadratic Functions

### Try This

Meanwhile, if you want to explore further what happens when you change the coefficients of a quadratic equation, the page at http://www.analyzemath.com/quadraticg/quadraticg.htm has an applet you can use. Click on the “Click here to start” button in section A, and then use the sliders to change the values of a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*}.

### Guidance

The graphs of quadratic functions are curved lines called parabolas. You don’t have to look hard to find parabolic shapes around you. Here are a few examples:

• The path that a ball or a rocket takes through the air.
• Water flowing out of a drinking fountain.
• The shape of a satellite dish.
• The shape of the mirror in car headlights or a flashlight.
• The cables in a suspension bridge.

#### Example A

Let’s see what a parabola looks like by graphing the simplest quadratic function, y=x2\begin{align*}y=x^2\end{align*}.

We’ll graph this function by making a table of values. Since the graph will be curved, we need to plot a fair number of points to make it accurate.

x\begin{align*}x\end{align*} y=x2\begin{align*}y = x^2\end{align*}
3\begin{align*}-3\end{align*} (3)2=9\begin{align*}(-3)^2 = 9\end{align*}
–2 (2)2=4\begin{align*}(-2)^2 = 4\end{align*}
–1 (1)2=1\begin{align*}(-1)^2 = 1\end{align*}
0 (0)2=0\begin{align*}(0)^2 = 0\end{align*}
1 (1)2=1\begin{align*}(1)^2 = 1\end{align*}
2 (2)2=4\begin{align*}(2)^2 = 4\end{align*}
3 (3)2=9\begin{align*}(3)^2 = 9\end{align*}

Here are the points plotted on a coordinate graph:

To draw the parabola, draw a smooth curve through all the points. (Do not connect the points with straight lines).

Let’s graph a few more examples.

#### Example B

Graph the following parabolas.

a) y=2x2+4x+1\begin{align*}y=2x^2 + 4x + 1\end{align*}

b) y=x2+3\begin{align*}y = -x^2 + 3\end{align*}

c) y=x28x+3\begin{align*}y = x^2 - 8x + 3\end{align*}

Solution

a) y=2x2+4x+1\begin{align*}y=2x^2 + 4x + 1\end{align*}

Make a table of values:

x\begin{align*}x\end{align*} y=2x2+4x+1\begin{align*}y = 2x^2 + 4x + 1\end{align*}
3\begin{align*}-3\end{align*} 2(3)2+4(3)+1=7\begin{align*}2(-3)^2 + 4(-3) + 1 = 7\end{align*}
–2 2(2)2+4(2)+1=1\begin{align*}2(-2)^2 + 4(-2) + 1 = 1\end{align*}
–1 2(1)2+4(1)+1=1\begin{align*}2(-1)^2 +4(-1) + 1 = -1\end{align*}
0 2(0)2+4(0)+1=1\begin{align*}2(0)^2 +4(0) + 1 = 1\end{align*}
1 2(1)2+4(1)+1=7\begin{align*}2(1)^2 +4(1) + 1 = 7\end{align*}
2 2(2)2+4(2)+1=17\begin{align*}2(2)^2 +4(2) + 1 = 17\end{align*}
3 2(3)2+4(3)+1=31\begin{align*}2(3)^2 +4(3) + 1 = 31\end{align*}

Notice that the last two points have very large y\begin{align*}y-\end{align*}values. Since we don’t want to make our y\begin{align*}y-\end{align*}scale too big, we’ll just skip graphing those two points. But we’ll plot the remaining points and join them with a smooth curve.

b) y=x2+3\begin{align*}y = -x^2 + 3\end{align*}

Make a table of values:

x\begin{align*}x\end{align*} y=x2+3\begin{align*}y = - x^2 + 3\end{align*}
3\begin{align*}-3\end{align*} (3)2+3=6\begin{align*}- (-3)^2 + 3 = -6\end{align*}
–2 (2)2+3=1\begin{align*}- (-2)^2 + 3 = -1\end{align*}
–1 (1)2+3=2\begin{align*}- (-1)^2 + 3 = 2\end{align*}
0 (0)2+3=3\begin{align*}- (0)^2 + 3 = 3\end{align*}
1 (1)2+3=2\begin{align*}- (1)^2 + 3 = 2\end{align*}
2 (2)2+3=1\begin{align*}- (2)^2 + 3 = -1\end{align*}
3 (3)2+3=6\begin{align*}-(3)^2 + 3 = -6\end{align*}

Plot the points and join them with a smooth curve.

Notice that this time we get an “upside down” parabola. That’s because our equation has a negative sign in front of the x2\begin{align*}x^2\end{align*} term. The sign of the coefficient of the x2\begin{align*}x^2\end{align*} term determines whether the parabola turns up or down: the parabola turns up if it’s positive and down if it’s negative.

c) y=x28x+3\begin{align*}y = x^2 - 8x + 3\end{align*}

Make a table of values:

x\begin{align*}x\end{align*} y=x28x+3\begin{align*}y = x^2 - 8x + 3\end{align*}
3\begin{align*}-3\end{align*} (3)28(3)+3=36\begin{align*}(-3)^2 - 8(-3) + 3 = 36\end{align*}
–2 (2)28(2)+3=23\begin{align*}(-2)^2 - 8(-2) + 3 = 23\end{align*}
–1 \begin{align*}(-1)^2 - 8(-1) + 3 = 12\end{align*}
0 \begin{align*}(0)^2 - 8(0) + 3 = 3\end{align*}
1 \begin{align*}(1)^2 - 8(1) + 3 = -4\end{align*}
2 \begin{align*}(2)^2 - 8(2) + 3 = -9\end{align*}
3 \begin{align*}(3)^2 - 8(3) + 3 = -12\end{align*}

Let’s not graph the first two points in the table since the values are so big. Plot the remaining points and join them with a smooth curve.

Wait—this doesn’t look like a parabola. What’s going on here?

Maybe if we graph more points, the curve will look more familiar. For negative values of \begin{align*}x\end{align*} it looks like the values of \begin{align*}y\end{align*} are just getting bigger and bigger, so let’s pick more positive values of \begin{align*}x\end{align*} beyond \begin{align*}x = 3\end{align*}.

\begin{align*}x\end{align*} \begin{align*}y = x^2 - 8x + 3\end{align*}
\begin{align*}-1\end{align*} \begin{align*}(-1)^2 - 8(-1) + 3 = 12\end{align*}
0 \begin{align*}(0)^2 - 8(0) + 3 = 3\end{align*}
1 \begin{align*}(1)^2 - 8(1) + 3 = -4\end{align*}
2 \begin{align*}(2)^2 - 8(2) + 3 = -9\end{align*}
3 \begin{align*}(3)^2 - 8(3) + 3 = -12\end{align*}
4 \begin{align*}(4)^2 - 8(4) + 3 = -13\end{align*}
5 \begin{align*}(5)^2 - 8(5) + 3 = -12\end{align*}
6 \begin{align*}(6)^2 - 8(6) + 3 = -9\end{align*}
7 \begin{align*}(7)^2 - 8(7) + 3 = -4\end{align*}
8 \begin{align*}(8)^2 - 8(8) + 3 = 3\end{align*}

Plot the points again and join them with a smooth curve.

Now we can see the familiar parabolic shape. And now we can see the drawback to graphing quadratics by making a table of values—if we don’t pick the right values, we won’t get to see the important parts of the graph.

In the next couple of lessons, we’ll find out how to graph quadratic equations more efficiently—but first we need to learn more about the properties of parabolas.

Compare Graphs of Quadratic Functions

The general form (or standard form) of a quadratic function is:

Here \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} are the coefficients. Remember, a coefficient is just a number (a constant term) that can go before a variable or appear alone.

Although the graph of a quadratic equation in standard form is always a parabola, the shape of the parabola depends on the values of the coefficients \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*}. Let’s explore some of the ways the coefficients can affect the graph.

Dilation

Changing the value of \begin{align*}a\end{align*} makes the graph “fatter” or “skinnier”. Let’s look at how graphs compare for different positive values of \begin{align*}a\end{align*}.

#### Example C

The plot on the left shows the graphs of \begin{align*}y=x^2\end{align*} and \begin{align*}y=3x^2\end{align*}. The plot on the right shows the graphs of \begin{align*}y=x^2\end{align*} and \begin{align*}y= \frac{1}{3}x^2\end{align*}.

Notice that the larger the value of \begin{align*}a\end{align*} is, the skinnier the graph is – for example, in the first plot, the graph of \begin{align*}y=3x^2\end{align*} is skinnier than the graph of \begin{align*}y=x^2\end{align*}. Also, the smaller \begin{align*}a\end{align*} is, the fatter the graph is – for example, in the second plot, the graph of \begin{align*}y= \frac{1}{3}x^2\end{align*} is fatter than the graph of \begin{align*}y=x^2\end{align*}. This might seem counterintuitive, but if you think about it, it should make sense. Let’s look at a table of values of these graphs and see if we can explain why this happens.

\begin{align*}x\end{align*} \begin{align*}y = x^2\end{align*} \begin{align*}y = 3x^2\end{align*} \begin{align*}y= \frac{1}{3}x^2\end{align*}
\begin{align*}-3\end{align*} \begin{align*}(-3)^2 = 9\end{align*} \begin{align*}3(-3)^2 = 27\end{align*} \begin{align*}\frac{(-3)^2}{3} = 3\end{align*}
–2 \begin{align*}(-2)^2 = 4\end{align*} \begin{align*}3(-2)^2 = 12\end{align*} \begin{align*}\frac{(-2)^2}{3} = \frac{4}{3}\end{align*}
–1 \begin{align*}(-1)^2 = 1\end{align*} \begin{align*}3(-1)^2 = 3\end{align*} \begin{align*}\frac{(-1)^2}{3} = \frac{1}{3}\end{align*}
0 \begin{align*}(0)^2 = 0\end{align*} \begin{align*}3(0)^2 = 0\end{align*} \begin{align*}\frac{(0)^2}{3} = 0\end{align*}
1 \begin{align*}(1)^2 = 1\end{align*} \begin{align*}3(1)^2 = 3\end{align*} \begin{align*}\frac{(1)^2}{3} = \frac{1}{3}\end{align*}
2 \begin{align*}(2)^2 = 4\end{align*} \begin{align*}3(2)^2 = 12\end{align*} \begin{align*}\frac{(2)^2}{3} = \frac{4}{3}\end{align*}
3 \begin{align*}(3)^2 = 9\end{align*} \begin{align*}3(3)^2 = 27\end{align*} \begin{align*}\frac{(3)^2}{3} = 3\end{align*}

From the table, you can see that the values of \begin{align*}y=3x^2\end{align*} are bigger than the values of \begin{align*}y=x^2\end{align*}. This is because each value of \begin{align*}y\end{align*} gets multiplied by 3. As a result the parabola will be skinnier because it grows three times faster than \begin{align*}y=x^2\end{align*}. On the other hand, you can see that the values of \begin{align*}y= \frac{1}{3}x^2\end{align*} are smaller than the values of \begin{align*}y=x^2\end{align*}, because each value of \begin{align*}y\end{align*} gets divided by 3. As a result the parabola will be fatter because it grows at one third the rate of \begin{align*}y=x^2\end{align*}.

Orientation

As the value of \begin{align*}a\end{align*} gets smaller and smaller, then, the parabola gets wider and flatter. What happens when \begin{align*}a\end{align*} gets all the way down to zero? What happens when it’s negative?

Well, when \begin{align*}a = 0\end{align*}, the \begin{align*}x^2\end{align*} term drops out of the equation entirely, so the equation becomes linear and the graph is just a straight line. For example, we just saw what happens to \begin{align*}y=ax^2\end{align*} when we change the value of \begin{align*}a\end{align*}; if we tried to graph \begin{align*}y=0x^2\end{align*}, we would just be graphing \begin{align*}y = 0\end{align*}, which would be a horizontal line.

So as \begin{align*}a\end{align*} gets smaller and smaller, the graph of \begin{align*}y=ax^2\end{align*} gets flattened all the way out into a horizontal line. Then, when \begin{align*}a\end{align*} becomes negative, the graph of \begin{align*}y=ax^2\end{align*} starts to curve again, only it curves downward instead of upward. This fits with what you’ve already learned: the graph opens upward if \begin{align*}a\end{align*} is positive and downward if \begin{align*}a\end{align*} is negative.

#### Example D

What do the graphs of \begin{align*}y=x^2\end{align*} and \begin{align*}y=- x^2\end{align*} look like?

Solution:

You can see that the parabola has the same shape in both graphs, but the graph of \begin{align*}y=x^2\end{align*} is right-side-up and the graph of \begin{align*}y=-x^2\end{align*} is upside-down.

Vertical Shifts

Changing the constant \begin{align*}c\end{align*} just shifts the parabola up or down.

#### Example E

What do the graphs of \begin{align*}y=x^2, y=x^2+1, y=x^2- 1, y=x^2 + 2,\end{align*} and \begin{align*}y=x^2-2\end{align*} look like?

Solution:

You can see that when \begin{align*}c\end{align*} is positive, the graph shifts up, and when \begin{align*}c\end{align*} is negative the graph shifts down; in either case, it shifts by \begin{align*}|c|\end{align*} units. In one of the later Concepts, we’ll learn about horizontal shift (i.e. moving to the right or to the left). Before we can do that, though, we need to learn how to rewrite quadratic equations in different forms - our objective for the next Concept.

Watch this video for help with the Examples above.

CK-12 Foundation: 1001 Graphs of Quadratic Functions

### Guided Practice

Graph the quadratic function, \begin{align*}y=-x^2+2\end{align*}.

Solution:

We’ll graph this function by making a table of values. Since the graph will be curved, we need to plot a fair number of points to make it accurate.

\begin{align*}x\end{align*} \begin{align*}y = x^2\end{align*}
\begin{align*}-3\end{align*} \begin{align*}-(-3)^2+2 = -7\end{align*}
–2 \begin{align*}-(-2)^2 +2= -2\end{align*}
–1 \begin{align*}-(-1)^2+2 = 1\end{align*}
0 \begin{align*}-(0)^2 +2 = 2\end{align*}
1 \begin{align*}-(1)^2 +2= 1\end{align*}
2 \begin{align*}-(2)^2 +2= -2\end{align*}
3 \begin{align*}-(3)^2 +2= -7\end{align*}

Plot the points and connect them with a smooth curve:

### Explore More

For 1-5, does the graph of the parabola turn up or down?

1. \begin{align*}y =-2x^2 - 2x -3\end{align*}
2. \begin{align*}y =3x^2\end{align*}
3. \begin{align*}y =16 - 4x^2\end{align*}
4. \begin{align*}y =-100 + 0.25x^2\end{align*}
5. \begin{align*}y =3x^2 - 2x - 4x^2 + 3\end{align*}

For 6-10, which parabola is wider?

1. \begin{align*}y = x^2\end{align*} or \begin{align*}y = 4x^2\end{align*}
2. \begin{align*}y = 2x^2 + 4\end{align*} or \begin{align*}y=\frac{1}{2} x^2 + 4\end{align*}
3. \begin{align*}y =-2x^2 - 2\end{align*} or \begin{align*}y = -x^2 - 2\end{align*}
4. \begin{align*}y = x^2 + 3x^2\end{align*} or \begin{align*}y = x^2 + 3\end{align*}
5. \begin{align*}y = -x^2\end{align*} or \begin{align*}y = \frac{1}{10}x^2\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 10.1.

### Vocabulary Language: English

Coefficient

Coefficient

A coefficient is the number in front of a variable.
Dilation

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.
domain

domain

The domain of a function is the set of $x$-values for which the function is defined.
Horizontal shift

Horizontal shift

A horizontal shift is the result of adding a constant term to the function inside the parentheses. A positive term results in a shift to the left and a negative term in a shift to the right.
Parabola

Parabola

A parabola is the characteristic shape of a quadratic function graph, resembling a "U".
quadratic function

quadratic function

A quadratic function is a function that can be written in the form $f(x)=ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.
standard form

standard form

The standard form of a quadratic function is $f(x)=ax^{2}+bx+c$.
Symmetry

Symmetry

A figure has symmetry if it can be transformed and still look the same.
Vertex

Vertex

The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward.
vertical axis

vertical axis

The vertical axis is also referred to as the $y$-axis of a coordinate graph. By convention, we graph the output variable on the $y$-axis.

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