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Wize High School Grade 11 Math Textbook > Quadratic Functions

Quadratic Functions

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Exploring Quadratic Relations

Paper Airplane Experiment

In this experiment, you will launch a paper airplane through the air, and record its height along its flight path!
Step 1: Create your own paper airplane (feel free to try out different airplane models!)

Step 2: Using your phone, tablet, or camera, film yourself launching this paper airplane through the air. Make sure to position your camera so that it captures the entire flight path in one frame (do not move the camera as you launch the airplane)

Step 3: Watch the video while periodically pausing frames to plot out some points of your plane's flight path. For example,
  • The plane is first launched at 0 seconds, and its height is 1.7m above the ground
  • At 0.4 seconds, the plane is 1.9m above the ground
  • At 1.2 seconds, the plane is at its highest point, which is 3.1m above the ground
  • ...
*Tip: To determine the height of the plane, you can either place a vertical ruler or measuring tape next to you when filming so you can identify the plane's height in the video, or use your height as reference, and measure/estimate the height of the plane in the video

Alternative Option

If you are unable to perform this experiment, feel free to use the data shown in the table below:

Create a scatter plot and sketch the curve of best fit for this experiment.

What are some special features of this relation?
  • Shape of the curve of best fit:
  • If we extend the graph beyond 0, into the negative time values, we notice the curve of best fit is symmetric.
  • Is the dependent variable increasing or decreasing?
  • The curve of best fit increases as before time reaches 1.5 seconds, then it decreases at the same rate.
  • If we extend the curve of best fit, what do we know about its intercepts?
  • The curve of best fit crosses the y-axis once, so there is one y-intercept. But if we extend the curve, we see that it crosses the x-axis twice, so there are two x-intercepts.
  • What do you notice about the first and second differences from the table of values?
  • The first differences are not constant, but the second differences are constant

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Quadratic Functions

Two variables are related and have a quadratic relation if it has the following properties:
  • The graph of the relation is a parabola -- a symmetric "U" shape
  • The table of value for this relation has a constant second difference that is not zero
  • If the second differences are negative, the parabola opens down
  • If the second differences are positive, the parabola opens up
  • The equation of the relation has degree 2

Quadratic relations can be modelled by quadratic functions of the form f(x)=ax2+bx+cf(x)=ax^2+bx+c, where a≠0a\neq0

Different Forms of the Quadratic Equation

A quadratic equation must have degree 2, but there are a few different possible forms:
  • Standard form: f(x)=ax2+bx+cf(x)=ax^2+bx+c
  • Factored form: f(x)=a(x−r)(x−s)f(x)=a(x-r)(x-s)
  • Vertex form: f(x)=a(x−h)2+kf(x)=a(x-h)^2+k
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Example: Quadratic Relation

Show that the following are quadratic relations. If the graph is not given, sketch the graph.

a)

The graph is a parabola ("U" shape). It is symmetrical along the y-axis, it is decreasing to the left of the y-axis and increasing to the right of the y-axis.
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b)

Looking at this table of values, you can see that the y-values appear to be symmetric, but let's determine the second differences to be sure that this is a quadratic relation.

Since the second differences are constant, this does represent a quadratic relation.

Here is the graph of the relation:

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c) y=−2(x+4)2−3y=-2(x+4)^2-3

Let's first expand and simplify the expression:
y=−2(x+4)2−3y=−2(x+4)(x+4)−3y=−2(x+4)(x+4)−3y=−2[(x)(x)+(x)(4)+(4)(x)+(4)(4)]−3y=−2[x2+4x+4x+16]−3y=−2[x2+8x+16]−3y=−2[x2+8x+16]−3y=(−2)(x2)+(−2)(8x)+(−2)(16)−3y=−2x2−16x−32−3y=−2x2−16x−35\begin{aligned} y&=-2(x+4)^2-3\\ y&=-2(x+4)(x+4)-3\\ y&=-2(\bcth{x}+\bct{4})(x+4)-3\\ y&=-2[\bcth{(x)}(x)+\bcth{(x)}(4)+\bct{(4)}(x)+\bct{(4)}(4)]-3\\ y&=-2[x^2+4x+4x+16]-3\\ y&=-2[x^2+8x+16]-3\\ y&=\bcfi{-2}[x^2+8x+16]-3\\ y&=\bcfi{(-2)}(x^2)+\bcfi{(-2)}(8x)+\bcfi{(-2)}(16)-3\\ y&=-2x^2-16x-32-3\\ y&=-2x^2-16x-35 \end{aligned}
We see that this equation is of the form y=ax2+bx+cy=ax^2+bx+c, it has degree 2. So, it is a quadratic relation.

Here's the graph for this relation:

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Practice: Quadratic Relation

i) Show that the following are quadratic relations
ii) If the graph is not given, sketch the graph representing the quadratic relation.

a)


b)


c) y=3(x−1)(x+2)y=3(x-1)(x+2)

Practice: Quadratic VS Linear Relations

For each of the following relations,

a) indicate the degree of the relation
b) state whether it represents a linear relation, quadratic relation, or neither.
y+x=3(x−2)−1y+x=3(x-2)-1

Practice: Different Forms of a Quadratic Function

Express the following quadratic functions in standard form (i.e. general form)
a) y=(3x−1)(2−5x)y=\left(3x-1\right)\left(2-5x\right)

b) f(x)=4(x−1)2−7f\left(x\right)=4\left(x-1\right)^2-7

Make sure to simplify your answer!

Practice: Quadratic Relations

Select all of the situations that can be modelled with a quadratic relation.