Graphing Parabolas Lesson

Up to this point, we have only graphed linear equations. As we move forward in math, we will learn how to graph equations/functions with more complex shapes. In this lesson, we will introduce the concept of graphing a quadratic equation. The graph of a quadratic equation is known as a "parabola". We will also learn about horizontal and vertical shifts. Let's begin by looking at the graph of the equation:
y = x²
To graph this equation, we want to set up a table of ordered pairs. With a line, we only needed two points. For a good sketch of a parabola, we need a lot more. In most cases, five points will suffice.

x	y	(x, y)
-2	4	(-2, 4)
-1	1	(-1, 1)
0	0	(0, 0)
1	1	(1, 1)
2	4	(2, 4)

Now, we can plot our points on the coordinate plane and draw a smooth curve through the points. When we work with parabolas, the vertex is the lowest or highest point on the graph. In our case, the vertex is the lowest point and occurs at the point (0,0). A vertical line drawn through the vertex is known as the axis of the parabola. A parabola is symmetric about its axis, this means we can fold our graph and the two curves will coincide. As a general rule:
f(x) = ax² + bx + c, a ≠0
The graph for a quadratic function is always a parabola with a vertical axis. To keep things simple in this lesson, we will use "y" in the place of "f(x)".

Vertical Shifts

Once we know the shape of y = x², we can use this graph to sketch other related graphs. Suppose we wanted to graph the following:
y = x² + 2
The 2 that is added to the right side of our original equation increases each y-value by 2 for the same x-value. In other words, the graph of y = x² + 2, will be the graph of y = x² shifted up by 2 units. This may not make sense at first, but let's look at a few ordered pairs using a table.

y = x2		y = x2 + 2
x	y	x	y
-2	4	-2	6
-1	1	-1	3
0	0	0	2
1	1	1	3
2	4	2	6

Let's look at these two parabolas on the same coordinate plane. We can see that our original graph has shifted two units up. In general, we can identify a vertical shift as follows:
y = x² + k
is a parabola with the same shape as:
y = x²
The difference is that y = x² + k is shifted:
up k units, if k is positive (k > 0)
down |k| units, if k is negative (k < 0)
The vertex for a parabola of this form occurs at (0,k). Let's look at a few examples.
Example 1: Identify the vertex and state the vertical shift.
y = x² - 11
We can rewrite this as:
y = x² + (-11)
Therefore, we know our k is (-11). This means our graph is shifted down 11 units. The vertex will occur at (0, -11).
Example 2: Identify the vertex and state the vertical shift.
y = x² + 8
Since k is 8, our graph is shifted up 8 units. The vertex will occur at (0, 8).

Horizontal Shifts

In addition to vertical shifts, we also have horizontal shifts. These are more challenging to understand since they are counterintuitive. Suppose we wanted to graph the following equation.
y = (x - 2)²
How would this compare to our original equation of:
y = x²
One might think this new equation is shifted 2 units to the left since we see a minus 2 inside of the parentheses, but the opposite is true. This equation is shifted 2 units to the right. Let's compare ordered pairs in our table.

y = x2		y = (x - 2)2
x	y	x	y
-2	4	-2	16
-1	1	-1	9
0	0	0	4
1	1	1	1
2	4	2	0

Again, this may be a bit hard to understand at first, but let's think about the ordered pair (-2,4) from the first equation, and (0, 4) from the second equation. Notice how the same y-value of 4 is produced from an x-value that is 2 units larger. In other words, for a given vertical position or y-value, the x-value will be 2 units larger or be 2 units to the right. This is what creates our horizontal shift of two units to the right. Let's look at the graphs side by side. In general, we can identify a horizontal shift as follows:
y = (x - h)²
is a parabola with the same shape as:
y = x²
The parabola is shifted
h units right if h is positive (h > 0)
|h| units left if h is negative (h < 0)
Note: Recall that minus a negative yields plus a positive:
For example:
3 - (-3) = 3 + 3 = 6
Note: The minus sign is given in the formula. This is a very common source of confusion.
We are only looking at the value of h.
In other words, we are moving right when subtracting away a positive and left when adding a positive.
The vertex will occur at (h,0)
Let's look at a few examples.
Example 3: Identify the vertex and state the horizontal shift.
y = (x - 3)²
We can identify h as 3, this means our graph will shift 3 units to the right. The vertex will occur at (3,0).
Example 4: Identify the vertex and state the horizontal shift.
y = (x + 9)²
We need to write this in the format of:
y = (x - h)²
y = (x - (-9))²
We can identify h as (-9), this means our graph will shift 9 units to the left. The vertex will occur at (-9,0).

Horizontal and Vertical Shifts

In most cases, we will have both a horizontal and a vertical shift. Suppose we looked at the following:
y = (x - 4)² - 12
We combine our strategies from the last two sections. We would see a shift to the right of 4 units and a shift down by 12 units. The vertex will occur at (h,k). In this case, our vertex will occur at (4, -12). Let's look at an example.
Example 5: Identify the vertex and state the horizontal and vertical shift.
y = (x + 7)² - 1
Let's rewrite our problem as:
y = (x - (-7))² + (-1)
This gives us an h-value of (-7) and a k-value of (-1). Our vertex will be at (-7,-1). The horizontal shift is 7 units left and the vertical shift is 1 unit down.

Graphing Downward Facing Parabolas

In some cases, we will see parabolas that face down instead of up. The same rules we listed above will apply. Let's graph
y = -x²
The negative coefficient on the squared term is what makes this parabola open down. The vertex (0,0) will now be the highest point, instead of the lowest. Let's suppose we wanted to graph:
y = -(x - 4)² + 5
We can shift the graph of y = -x², 4 units right and 5 units up. The vertex or highest point will occur at (4,5).

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Skills Check: