Wize High School Grade 11 Math Textbook > Quadratic Functions

Properties of Quadratic Graphs

Factored Form of a Quadratic EquationNext Section

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Properties of Quadratic Graphs
We are often interested in certain properties of the graph of a quadratic relation:

The axis of symmetry
This is an equation of the form      x = number     
 
It represents a vertical line that separates the parabola into two identical parts

The direction of the opening
If the parabola looks like a right side up U, then it is opening up
If the parabola looks like an upside down U, then it is opening down

The vertex
This is a point where the parabola turns around from increasing to decreasing or decreasing to increasing
If the parabola is opening up, then the vertex is at the very bottom, the y-coordinate represents the minimum value
If the parabola is opening down, then the vertex is at the very top, the y-coordinate represents the maximum value

The y\bcf{y}y-intercept
This is where the parabola meets the y-axis
A parabola will have exactly 1 y-intercept

The x\bcf{x}x-intercepts,
This is where the parabola meets the x-axis
This is also known as the      zeros     
 or the      roots     
A parabola can have       0, 1, or 2     
 possible x-intercepts

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Example: Properties of Parabolas (Graphs of Quadratic Relations)
For each of the following quadratic relations, identify
i) the axis of symmetry
ii) the direction of the opening
iii) the vertex
iv) the y-intercept
v) the x-intercept(s)

a) y=x2+2x+4y=x^2+2x+4y=x2+2x+4

axis of symmetryx=−1direction of openingupvertex(−1,3)maximum or minimum valuemin=3y-intercept(0,4)x-intercept(s)None\begin{array}{|c|c|}
\hline
\text{axis of symmetry}&x=-1\\

\hline
\text{direction of opening}&\text{up}\\

\hline
\text{vertex}&(-1,3)\\

\hline
\text{maximum or minimum value}&\text{min}=3\\

\hline
\text{y-intercept}&(0,4)\\

\hline
\text{x-intercept(s)}&\text{None}\\
\hline

\end{array}axis of symmetrydirection of openingvertexmaximum or minimum valuey-interceptx-intercept(s)​x=−1up(−1,3)min=3(0,4)None​​

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b) y=x2−4x+3y=x^2-4x+3
y=x2−4x+3

axis of symmetryx=2direction of openingupvertex(2,−1)maximum or minimum valuemin=−1y-intercept(0,3)x-intercept(s)(1,0) & (3,0)\begin{array}{|c|c|}
\hline
\text{axis of symmetry}&x=2\\

\hline
\text{direction of opening}&\text{up}\\

\hline
\text{vertex}&(2,-1)\\

\hline
\text{maximum or minimum value}&\text{min}=-1\\

\hline
\text{y-intercept}&(0,3)\\

\hline
\text{x-intercept(s)}&(1,0)~\&~(3,0)\\
\hline

\end{array}axis of symmetrydirection of openingvertexmaximum or minimum valuey-interceptx-intercept(s)​x=2up(2,−1)min=−1(0,3)(1,0) & (3,0)​​

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c) y=−x2+4y=-x^2+4y=−x2+4

axis of symmetryx=0direction of openingdownvertex(0,4)maximum or minimum valuemax=4y-intercept(0,4)x-intercept(s)(−2,0) & (2,0)\begin{array}{|c|c|}
\hline
\text{axis of symmetry}&x=0\\

\hline
\text{direction of opening}&\text{down}\\

\hline
\text{vertex}&(0,4)\\

\hline
\text{maximum or minimum value}&\text{max}=4\\


\hline
\text{y-intercept}&(0,4)\\

\hline
\text{x-intercept(s)}&(-2,0)~\&~(2,0)\\
\hline

\end{array}axis of symmetrydirection of openingvertexmaximum or minimum valuey-interceptx-intercept(s)​x=0down(0,4)max=4(0,4)(−2,0) & (2,0)​​

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d) y=−2x2+4x−2y=-2x^2+4
x-2y=−2x2+4x−2

axis of symmetryx=1direction of openingdownvertex(1,0)maximum or minimum valuemax=0y-intercept(0,−2)x-intercept(s)(1,0)\begin{array}{|c|c|}
\hline
\text{axis of symmetry}&x=1\\

\hline
\text{direction of opening}&\text{down}\\

\hline
\text{vertex}&(1,0)\\

\hline
\text{maximum or minimum value}&\text{max}=0\\


\hline
\text{y-intercept}&(0,-2)\\

\hline
\text{x-intercept(s)}&(1,0)\\
\hline

\end{array}axis of symmetrydirection of openingvertexmaximum or minimum valuey-interceptx-intercept(s)​x=1down(1,0)max=0(0,−2)(1,0)​​

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Summary
Wize Tip
The direction of opening
If the aaa value is negative, then the parabola opens      down     
If the aaa value is positive, then the parabola opens      up     

If the vertex has coordinates (h,k)(h, k)(h,k)
The axis of symmetry is      x = h     
If the opening is up, the      min     
 value is      k     
 
If the opening is down, the      max     
 value is      k     

x-intercepts (zeros or roots)
If the parabola opens up, and the vertex has a positive y-coordinate (k>0k>0k>0), how many x-intercepts does it have?      0     
If the parabola has a vertex that is on the x-axis, how many x-intercepts does it have?      1     
If the parabola opens up, and the vertex has a negative y-coordinate (k<0k<0k<0), how many x-intercepts does it have?      2     
If the parabola opens down, and the vertex has a positive y-coordinate (k>0k>0k>0), how many x-intercepts does it have?      2     

Practice: Properties of Quadratic Graphs
Given the quadratic equation y=−x2+4x+5y=-x^2+4x+5y=−x2+4x+5,
a) create a table of values.

b) graph the parabola.

c) Identify the vertex.

d) Identify the axis of symmetry.

e) Identify the direction of the opening.

f) Identify the y-intercept.

g) Identify the x-intercept(s). 
 If there is no x-intercept, enter DNE
If there is more than one x-intercept, enter them in the form (x,0), (x,0)(x,0),~(x,0)(x,0), (x,0), put the one with the smaller x value first

h) Identify the max or min value.

a) and b) - Check your table of values and graph of the parabola against the solutions

c) Enter the vertex in the form

(x,y)

d) Enter the axis of symmetry in the form

x=number

e) Enter up or down for the direction of the parabola opening

f) Enter the y-intercept in the form

(0,y)

g) Enter the x-intercept(s) in the form

(x,0)

. If there is no x-intercept, enter DNE. If there is more than one x-intercept, enter them in the form

(x,0),~(x,0)

, put the one with the smaller x value first.

h) Enter your answer in the form

max=number

min=number

I don't know

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Practice: Sketching Quadratic Graphs
Sketch a parabola with the following information:
a) x-intercepts of(3,0)(3,0)(3,0) and (7,0)(7,0)(7,0), and a maximum value of 5.
b) y-intercept of (0,5)(0,5)(0,5), an axis of symmetry of x=−1x=-1x=−1, and a minimum value of −2-2−2.

I have answered this question

Practice: Axis of Symmetry & Vertex
A parabola passes through the points (−2,1)(-2,1)(−2,1) and (0,1)(0,1)(0,1).

a) Determine the x-coordinate of the vertex.

b) If this quadratic equation is y=2x2+4x+1y=2x^2+4x+1y=2x2+4x+1, determine the y-coordinate of the vertex.

a) The x-coordinate of the vertex is

x=

b) The y-coordinate of the vertex is

y=

I don't know

Practice: Rocket Launch
A small toy rocket is launched from a platform slightly above the ground. It's path can be modelled by the equation h=−(t+1)(t−3)h=-(t+1)(t-3)h=−(t+1)(t−3), where hhh represents the height of the rocket in meters and ttt represents the number of seconds after the rocket was launched.

a) Expand and simplify the equation to show that it has degree 2.

b) Create a table of values and sketch the parabola.

c) What is the maximum height of this rocket, and when does it reach this height?

d) What is the height of the platform the rocket was launched from?

e) When does the rocket hit the ground?

a) Enter the simplified equation in the form

h=

b) Check your table of values and graph against the solutions

c) Enter the maximum height of the rocket in meters, do not include units

c) Enter the time (in seconds) when the rocket reaches its maximum height, do not include units

d) Enter the height of the launching platform in meters, do not include units

e) Enter the time (in seconds) when the rocket hits the ground, do not include units

I don't know

Practice: Revenue
A small high school musical committee wants to determine the optimal price to sell its tickets in order to maximize revenue. The current ticket price is $10, and 20 people will purchase a ticket at this price. Let RRR represent the revenue and xxx represent the number of $1 increases to the ticket price. Then the revenue from this school play is modelled by the equation R=(10+x)(20−x)R=(10+x)(20-x)R=(10+x)(20−x).

a) Create a table of values and sketch this parabola.

b) Determine the maximum revenue and at what ticket price this occurs.

c) At what ticket price will the revenue be $0?

a) Check the table of values and graph against the solutions

b) Enter the value of the maximum revenue (in $), do not include units

b) Enter the ticket price (in $) that results in the maximum revenue, do not include units

c) Enter the ticket price (in

) that results in a revenue of

0, do not include units

I don't know

Wize High School Grade 11 Math Textbook > Quadratic Functions

Properties of Quadratic Graphs

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Properties of Quadratic Graphs

Example: Properties of Parabolas (Graphs of Quadratic Relations)

Summary

Practice: Properties of Quadratic Graphs

Practice: Sketching Quadratic Graphs

Practice: Axis of Symmetry & Vertex

Practice: Rocket Launch

Practice: Revenue