Quadratic Relationships
Unit 1 - Graphing Vertex Form
Learning Goals
By the end of this unit you should be able to:
- Understand what each variable means in a vertex form equation (i.e. a, h, and k)
- Locate/describe the characteristics of a parabola (i.e. vertex, axis of symmetry, etc.)
- Describe the transformations of, and graph, quadratic relations in vertex form
Summary
The equation of a parabola in Vertex Form is y = a(x - h)² + k.
You can use this equation to solve quadratic relations.
The Variables:
- The value of a tells you the direction of opening and whether the parabola is compressed or stretched. If a > 0, the parabola opens upwards and if a < 0, the parabola opens downwards. If -1 < a < 1, the parabola will be vertically compressed and if a > 1 or a < -1, the parabola will be vertically stretched.
- The value of h tells you how much the parabola has translated horizontally from the origin. If h > 0, the vertex moves to the right h units and if h < 0, the vertex moves to the left h units.
- The value of k tells you how much the parabola has translated vertically from the origin. If k > 0, the vertex moves up k units and if k < 0, the vertex moves down k units.
The Vertex is the highest or lowest point on a parabola. It is the h and k value and can be written as (h,k) or (x,y).
The Axis of Symmetry is an imaginary line that goes straight through the vertex and divides the parabola into two equal halves. It can be written as (x = h).
The Optimal Value is the value of the y co-ordinate of the vertex. It can be written as (y = k).
To find the y-intercept, set x=0 in the equation and solve for y. And for x-intercept, set y=0 and solve for x.
Examples of Graphs
Parabola Opening Upwards
This is a positive parabola, because the value of a is positive, so it opens upward. An example of a vertex form equation of a positive parabola would be
y = 2(x - 1)² + 3.
Parabola Opening Downwards
This is a negative parabola, because the value of a is negative, so it opens downward. An example of a vertex form equation of a negative parabola would be
y = -1(x - 2)² + 3.
Parabola with Two x-intercepts
Like the name suggests, this parabola has 2 x-intercepts. That is because 2 points of the parabola are crossing the x axis. (Not all parabolas have 2 x-intercepts). An example of a vertex form equation of a this parabola would be
y = (x - 1)² -2.
Example of a Word Problem
The flight path of a firework is modeled by the relation h = −5(t − 5)² + 127, where h is the height, in metres, of the firework above the ground and t is the time, in seconds, since the firework was fired.
= −5(0 − 5)² + 127
= −5(− 5)² + 127
= −5(25) + 127
= −125 + 127
h = 2 m
- What is the maximum height reached by the firework?
- How many seconds after it was fired does the firework reach this height?
- How high was the firework above the ground when it was fired?
Solutions:
- The maximum height reached by the firework is 127 m. That is because in the given equation, 127 is the k value and k represents the maximum (or minimum) height of a parabola.
- The firework reaches this height after 5 seconds. That is because in the given equation, 5 (in the brackets) is the h value and h represents the horizontal distance of the maximum (or minimum) point on a parabola. The reason why 5 is positive, even though it is negative inside the brackets, is because when a number comes out of brackets, the sign in front of it switches.
- The firework was 2 m above the ground when it was fired. To find that out, you have to find the y-intercept because that is the where the parabola started. So substitute 0 into the equation as the t (x) value and solve for h (y).
= −5(0 − 5)² + 127
= −5(− 5)² + 127
= −5(25) + 127
= −125 + 127
h = 2 m
Video of Graphing Vertex Form Using Transformations
Quick Way of Graphing a Quadratic Function in Vertex Form
Graphing Quadratic Equations Using Transformations