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

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.


  1. What is the maximum height reached by the firework?
  2. How many seconds after it was fired does the firework reach this height?
  3. How high was the firework above the ground when it was fired?

Solutions:


  1. 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.
  2. 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.
  3. 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).
h = −5(t − 5)² + 127
= −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

Unit 2 - Factored Form

Learning Goals

By the end of this unit you should be able to:


  • Expand a binomial multiplied by a binomial
  • Common factor a polynomial
  • Factor complex trinomials and find 2 x-intercepts and vertex
  • Factor perfect square trinomials and differences of squares

Summary

An equation in Factored Form looks like this: y = a(x - r) (x - s)


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
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Example of a Word Problem

The path of a toy rocket is defined by the relation y = -3x² + 11x + 4, where x is the horizontal distance, in metres, travelled and y is the height, in metres, above the ground.


  1. Determine the zeros of the relation.
  2. How far has the rocket travelled horizontally when it lands on the ground?
  3. What is the maximum height of the rocket above the ground, to the nearest hundredth of a metre?

Solutions:

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How to Factor any Quadratic Equation
Factoring Special Products

Unit 3 - Standard Form

Learning Goals

By the end of this unit you should be able to:


  • 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

An equation in Factored Form looks like this: y = ax² + bx + c


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

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Completing the Square Example

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Example of a Word Problem

The path of a basketball after it is thrown in the air is given by the following equation:

h = -0.25d² + 2d + 1.5, where h is the height and d is the horizontal distance in metres.


  1. What is the initial height of the basketball?
  2. What is the maximum height reached by the basketball and at what horizontal distance does this occur at?


Solutions:

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Reflection on Quadratics

Overall, I have learned a lot about the various forms of quadratic relations. I now know, how to solve and graph quadratic relations in vertex, factored, and standard forms. This will be very useful to me, as I continue on my academic path of education. The vertex form can provide the vertex to me (h and k values). Factored form provides me with the x-intercepts/solutions (r and s values). And standard form can provide the y-intercept, as well as the vertex (completing the square) and x-intercepts (quadratic formula). All of the forms connect in some way, they all can help graph a quadratic relation. That is because once you have the vertex, x-intercepts, or even the y-intercept and a reflected point... you can just plot them down on a graph and draw your parabola. I enjoyed learning about quadratics and how they are also present in our world (roller coaster, bridges, etc.).

Example of Assesment

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Created by Neel Patel