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Wize High School Algebra I Textbook (Common Core) > Quadratic Functions

Quadratic Relations

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Quadratic Relations

Recall - Linear Relations

Linear relations is a special types of relationship between two variables
  • When the value of the independent variable changes, the value of the dependent variable changes by a proporationate amount
  • In a table of values representing a linear relation, we will see constant first differences
  • In an equation representing a linear relation, we will see a degree of 1 for both independent and dependent variables
  • In a graph represending a linear relation, we will see a straight line

But what if a relation is not linear, are there other types of special relations?

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Paper Airplane Experiment

A paper plane is launched into the air. The following table shows the height of this plane at different points in time. 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 Relation

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
  • The equation of the relation has degree 2

Different Forms of the Quadratic Equation

A quadratic equation must have degree 2, but there are a few different possible forms:
  • Standard form: y=ax2+bx+cy=ax^2+bx+c
  • Factored form: y=a(x−r)(x−s)y=a(x-r)(x-s)
  • Vertex form: y=a(x−h)2+ky=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:

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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: Quadratic Relations

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