Remember that a quadratic function has an equation of the form $y=ax^2+bx+c$y=ax2+bx+c, where $b$b and $c$c can be any number and $a$a can be any number except for zero. For example, the equations $y=2x^2$y=2x2 and $y=x^2-3x+4$y=x2−3x+4 are both quadratic.

In this section we are going to use technology to help us find the graphs of parabolas from their equations, and then look at the features of those graphs.

Using technology

There are many ways of using technology to obtain graphs from equations. Some of them are listed here:

There are a number of websites that can create graphs given equations, such as www.geogebra.org and www.wolframalpha.com.
Some calculators (often referred to as graphing calculators) have the ability to plot graphs.
There are many apps available for smartphones that can plot graphs from equations.

Note that the graphs displayed in this lesson have been created using Geogebra.

Here are five parabolas. Take a look at the shapes - what similarities are there? What differences do you notice?

Notice that:

they are all symmetric about the $y$y-axis. So the axis of symmetry for all these parabolas is $x=0$x=0.
they all open upwards. So they are all concave up (have positive concavity).

But there are some differences too:

They have different $y$y-intercepts.
Some of them have no $x$x-intercepts, some have one $x$x-intercept, and some have two.

Another set of five parabolas has been graphed below. What similarities and differences are there for these parabolas?

Notice that:

they all have the same axis of symmetry. They are all symmetric about the line $x=2$x=2.
they all open upwards. So they are all concave up (have positive concavity).
they all have the same turning point (or vertex) at $\left(2,0\right)$(2,0).

Here is a final set of parabolas:

Notice that the turning point of each parabola lies on the x-axis. This also means that they all have one $x$x-intercept, where the curve bounces off the $x$x-axis.

Three of the parabolas are concave up, so they have minimum turning points. Two are concave down, so they have maximum turning points.

Practice questions

Question 1

Using technology, sketch the graph of $y=2x^2$y=2x2.

Then answer the following questions:

Does this parabola have a maximum or a minimum value?
Maximum
A
Minimum
B
What is the minimum $y$y value on the graph?
Which of the following statements is true?
The parabola does not have an axis of symmetry.
A
The parabola has an axis of symmetry along the $y$y-axis of the graph.
B
The parabola has an axis of symmetry that is vertical, but not on the $y$y-axis.
C
Now use technology to sketch the graph of $y=x^2$y=x2 on the same axes.

Which of the following statements is true?
The graph of $y=2x^2$y=2x2 has the same minimum value as $y=x^2$y=x2, but a different axis of symmetry.
A
The graph of $y=2x^2$y=2x2 has the same axis of symmetry as $y=x^2$y=x2, but a different minimum value.
B
The graphs of $y=2x^2$y=2x2 and $y=x^2$y=x2 have the same minimum value and axis of symmetry, but the graph of $y=2x^2$y=2x2 is narrower.
C
The graphs of $y=2x^2$y=2x2 and $y=x^2$y=x2 have the same minimum value and axis of symmetry, but the graph of $y=2x^2$y=2x2 is wider.
D

Question 2

Using technology, sketch the graph of $y=-x^2+3$y=−x2+3.

Then answer the following questions:

Does this parabola have a maximum or a minimum value?
Maximum
A
Minimum
B
What is the maximum $y$y value on the graph?
Which of the following statements is true?
The parabola has an axis of symmetry along the $y$y-axis of the graph.
A
The parabola has an axis of symmetry that is vertical, but not on the $y$y-axis.
B
The parabola does not have an axis of symmetry.
C
Now use technology to sketch the graph of $y=-x^2$y=−x2 on the same axes.

Which of the following statements is true?
The graph of $y=-x^2+3$y=−x2+3 has the same axis of symmetry as $y=-x^2$y=−x2, but a different maximum value.
A
The graphs of $y=-x^2+3$y=−x2+3 and $y=-x^2$y=−x2 have the same maximum value and axis of symmetry, but the graph of $y=-x^2+3$y=−x2+3 is wider.
B
The graph of $y=-x^2+3$y=−x2+3 has the same maximum value as $y=-x^2$y=−x2, but a different axis of symmetry.
C
The graphs of $y=-x^2+3$y=−x2+3 and $y=-x^2$y=−x2 have the same maximum value and axis of symmetry, but the graph of $y=-x^2+3$y=−x2+3 is narrower.
D

Question 3

Consider the quadratic equations $y=\frac{1}{2}x^2+2$y=12x2+2 and $y=\frac{1}{2}x^2-2$y=12x2−2.

Using technology, sketch the two parabolas on the same set of axes. Then answer the following questions:

Do these parabolas each have a minimum value or a maximum value?
Maximum
A
Minimum
B
Compare the two minimum values. Which parabola has the lower minimum value?
The graph of the quadratic $y=\frac{1}{2}x^2+2$y=12x2+2.
A
The graph of the quadratic $y=\frac{1}{2}x^2-2$y=12x2−2.
B
Which of the following statements is true?
Both parabolas cross the $x$x-axis twice.
A
Neither parabola crosses the $x$x-axis.
B
The graph of $y=\frac{1}{2}x^2+2$y=12x2+2 crosses the $x$x-axis twice, while the graph of $y=\frac{1}{2}x^2-2$y=12x2−2 does not cross the $x$x-axis.
C
The graph of $y=\frac{1}{2}x^2+2$y=12x2+2 does not cross the $x$x-axis, while the graph of $y=\frac{1}{2}x^2-2$y=12x2−2 crosses the $x$x-axis twice.
D
What are the $x$x values of the points where the graph of $y=\frac{1}{2}x^2-2$y=12x2−2 crosses the $x$x-axis?

Write both values on the same line, separated by a comma.

Outcomes

MS2-12-1

uses detailed algebraic and graphical techniques to critically evaluate and construct arguments in a range of familiar and unfamiliar contexts

MS2-12-6

solves problems by representing the relationships between changing quantities in algebraic and graphical forms

2.02 Graphing quadratic functions with technology