Vertex Form of a Parabola

The Vertex Form of a Parabola

The formula for the vertex form of a parabola is:
f(x) = a(x - h)2 + k
where: a = vertical stretch or shrink of the parabola
and (h, k) are the (x, y) coordinates of the vertex of the parabola.
h = the x-coordinate and k = the y-coordinate

A Parabola is a U-shaped graph that is vertically symmetrical about a line that intersects the vertex of the graph. parabola

A parabola can be rotated and face either up, or down, but never sideways. rotated p

Parabolas occur naturally in the real world when considering the flight of an object moving forward while in the air. A few examples are below.
angry birds angrybirds basketball baseball
This video demonstrates how pilots use parabolas in flight to produce zero-gravity situations:

Let's begin by learning how to substitute the values of a, h, and k into the formula

Given the values of a, h, and k, fill in the formula.
Example 1: a = 3, h = 2, k = 4
Formula: f(x) = a(x - h)2 + k
Solution:
f(x) = 3(x - 2)2 + 4

Example 2: a = 2, h = 5, k = 7
Formula: f(x) = a(x - h)2 + k
Solution:
f(x) = 2(x - 5)2 + 7

Example 3: a = 1, h = -4, k = -3
Formula: f(x) = a(x - h)2 + k
Solution:
f(x) = (x + 4)2 - 3

Example 4: a = -3, h = -1, k = -6
Formula: f(x) = a(x - h)2 + k
Solution:
f(x) = -3(x + 1)2 - 6

The Axis of Symmetry of a Parabola

Every parabola has an axis of symmetry that crosses through the vertex. The axis is a vertical line and therefore has an equation x = h.
Given each parabola, state the equation of the axis of symmetry.

Example 1:
give equation Solution:
First, we need to locate the vertex and write its location as an ordered pair. Since the x-coordinate is 4 and the y-coordinate is -3, the vertex is the ordered pair (4, -3). This implies that the vertex lies along the vertical line
x = 4.
Answer: x = 4 give equation

Example 2:
give equation Solution:
First, we need to locate the vertex and write its location as an ordered pair. Since the x-coordinate is -5 and the y-coordinate is 2, the vertex is the ordered pair (-5, 2). This implies that the vertex lies along the vertical line
x = -5.
Answer: x = -5 give equation

Example 3:
give equation Solution:
First, we need to locate the vertex and write its location as an ordered pair. Since the x-coordinate is 3 and the y-coordinate is 8, the vertex is the ordered pair (3, 8). This implies that the vertex lies along the vertical line
x = 3.
Answer: x = 3 give equation

The Maximum and Minimum of a Parabola

Parabolas may open upwards or downwards. Depending on the orientation of the graph, the vertex can be a maximum point or a minimum point.
  1. When a parabola faces upwards, the vertex is the lowest point of the graph. It is called a minimum because no part of the graph will go lower than the vertex. up graph
  2. When a parabola faces downwards, the vertex is the highest most point of the graph. There will not be any points higher than the vertex. Therefore, it is called a maximum. up graph

The y-Intercept of a Parabola
The y-intercept of a graph is the location where the graph crosses the y-axis. It is written as an ordered pair (0, y). The x-coordinate is always zero when you have the y-intercept. Using substitution of x = 0 into the equation and using PEMDAS to simplify the equation will give you the y-intercept.

Example 1:
Calculate the y-intercept of the parabola f(x) = 2(x - 3)2 + 6
Solution:
Substitute x = 0 into the equation then simplify.
f(x) = 2(0 - 3)2 + 6
f(x) = 2(-3)2 + 6
f(x) = 18 + 6
f(x) = 24
f(x) = 24 means y = 24 since f(x) and y mean the same thing. The y-intercept of the graph is at (0, 24).

Example 2:
Calculate the y-intercept of the parabola f(x) = 4(x - 2)2 + 5
Solution:
Substitute x = 0 into the equation then simplify.
f(x) = 4(0 - 2)2 + 5
f(x) = 4(-2)2 + 5
f(x) = 16 + 5
f(x) = 21
f(x) = 21 means y = 21 since f(x) and y mean the same thing. The y-intercept of the graph is at (0, 21).

Example 3:
Calculate the y-intercept of the parabola f(x) = -3(x + 1)2 - 6
Solution:
Substitute x = 0 into the equation then simplify.
f(x) = -3(0 + 1)2 - 6
f(x) = -3(1)2 - 6
f(x) = -3 - 6
f(x) = -9
f(x) = -9 means y = -9 since f(x) and y mean the same thing. The y-intercept of the graph is at (0, -9).

Using the vertex and the y-intercept, you can create a "rough-draft" of the graph of the parabola. Additional points on the graph can be found by substituting different values of x in to the equation and calculating the corresponding values of y. Turning the x-and y-values into ordered pairs (x, y) and plotting them on the graph will give you a sketch of the graph with increased accuracy. This process is covered in the "Graphing Parabolas" module.

Does the Parabola open upwards or downwards?
The value of the variable "a" in the formula f(x) = a(x - h)2 + k tells you whether the graph opens upwards or downwards.
  1. a is positive: the graph opens upwards. up
  2. a is negative: the graph opens downwards. down

Example 1: Determine whether the graph opens upwards or downwards.
f(x) = -5(x - 3)2 + 2
Solution: the value of "a" is -5. The graph opens downwards.

Example 2: Determine whether the graph opens upwards or downwards.
f(x) = 3(x + 9)2 + 7
Solution: the value of "a" is +3. The graph opens upwards.

In this lesson, you have learned a lot about the basics of a Parabola in vertex form. Each of these pieces come together to help you make a sketch of its graph. Each of the properties needs to be memorized.

  1. Formula f(x) = a(x - h)2 + k
  2. The Parabola always produces a "U"-shaped graph which can open upwards or downwards depending on the value of "a".
  3. Given the values of a, h, and k, fill them into the formula to produce the equation of the graph
  4. Locating the Axis of Symmetry of a parabola
  5. Determining the maximum or minimum of a parabola
  6. Calculating the y-intercept of a parabola