Quadratic Relationships
Graphing Vertex Form, Factored Form, and Standard Form
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.
Example 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. |
Parabola Opening Upwards
y = 2(x - 1)² + 3.
Parabola Opening Downwards
y = -1(x - 2)² + 3.
Example of a Word Problem
- 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
Unit 2 - Factored Form
Summary
The Variables:
- The value of a gives you the shape and direction of opening
- The value of r and s give you the x-intercepts
Axis of symmetry, AOS: x = (r + s) / 2 -- Sub this x value into the original equation to find the optimal value
To find the y-intercept, set x = 0 and solve for y
Types of Factoring:
- Greatest Common Factor
- Simple Trinomial factoring (a = 1)
- Complex Trinomial factoring
- Special case - Difference of squares
- Special case – Perfect square
Example of a Word Problem
- Determine the zeros of the relation.
- How far has the rocket travelled horizontally when it lands on the ground?
- What is the maximum height of the rocket above the ground, to the nearest hundredth of a metre?
Solutions:
Unit 3 - Standard Form
Learning Goals
- determine max/min values of a quadratic relation using zeroes and symmetry
- convert a quadratic relation from standard form to vertex form (complete the square)
- solve quadratic equations using the quadratic formula
- solve real-world problems involving max/min
Summary
The Variables:
- the value of a gives you the shape and direction of opening
- the value of c is the y-intercept
To get the x-intercepts, solve using the quadratic formula.
The number inside the square root (b² - 4ac) of the quadratic formula is called the Discriminant (D). It helps us tell how many solutions the quadratic equation will have without having to use the whole formula.
When D < 0, there will be no solutions (x-intercepts). When D > 0, there will be 2 solutions. And when D = 0, there will be only 1 solution.
Use the completing the square method to convert standard form into vertex form.
Quadratic Formula Example
Completing the Square Example
Example of a Word Problem
h = -0.25d² + 2d + 1.5, where h is the height and d is the horizontal distance in metres.
- What is the initial height of the basketball?
- What is the maximum height reached by the basketball and at what horizontal distance does this occur at?
Solutions: