Solving Quadratic Equations with Graphs

Slideshow 31, MathematicsMr. Richard Sasaki, Room 307

Objectives

• Recall how to draw graphs with linear equations

• Be able to solve simultaneous equations graphically with both linear and quadratic lines

• Be able to do this with real world examples

Linear Graphs

First let’s have a review on how to draw linear graphs.

Linear graphs are in the form .𝑦=𝑎𝑥+𝑏They are straight lines and much easier to draw than quadratic graphs.ExampleDraw the graph .

Answers

Solving Simultaneous Equations Graphically

If we have a pair of simultaneous equations, do you remember how to solve them graphically?ExampleSolve the simultaneous equations and with the use of a graph.First draw both of the lines.The solution is simply the intersection. (2 ,3)So we get .𝑥=2 , 𝑦=3

Answers

𝑥=−2 , 𝑦=−4𝑥=3 , 𝑦=−3𝑥=2 , 𝑦=7

𝑥=−1 , 𝑦=2𝑥=−3 , 𝑦=−8𝑥=2 , 𝑦=7

Where’s the vertex?(0 ,−2)

Quadratic Simultaneous Equations

Let’s solve some simultaneous equations with quadratics like in Grade 8, but with graphs.ExampleSolve the simultaneous equations and graphically below:

11

The parabola allows for up to two intersections:

𝑥=−1 , 𝑦=−1𝑥=3 , 𝑦=7

That’s our two pairs of solutions!

Answers - Easy

𝑥=−1 , 𝑦=−3𝑥=3 , 𝑦=5

𝑥=−3 , 𝑦=1𝑥=0 , 𝑦=4

𝑥=0 , 𝑦=−2𝑥=5 , 𝑦=3

𝑥=−3 , 𝑦=0𝑥=2 , 𝑦=5

Answers - Medium

𝑥=1 , 𝑦=2𝑥=4 , 𝑦=8

𝑥=−2 , 𝑦=−2𝑥=2 , 𝑦=6

𝑥=−2 , 𝑦=2𝑥=−5 , 𝑦=8

𝑥=0 , 𝑦=5𝑥=2 , 𝑦=5

Answers - Hard

𝑥=−4 , 𝑦=−3𝑥=−1 , 𝑦=3

𝑥=−9 , 𝑦=−7𝑥=−6 , 𝑦=−4

𝑥=−6 , 𝑦=0𝑥=−2 , 𝑦=8

𝑥=0 , 𝑦=−6𝑥=4 , 𝑦=2

Real World Equations and Graphs Now we will look at some non-simultaneous real world problems.ExampleThe driver of a car travelling downhill on a road applies the brakes. The speed of the car (in ) is equal to where is the number of seconds after starting to brake.First, we need to draw the graph . (We don’t need negative values.)

𝑦=−4 𝑥2+12𝑥+80𝑦−80=−4(𝑥¿¿2−3𝑥+

94−94)¿

𝑦=−4 (𝑥− 32 )2

+89Vertex:

Real World Equations and Graphs Now we need to place that vertex on the grid and draw the line. Vertex: 41112

85

73

120

53

1

2825

How fast was the car travelling when the brakes were applied?80𝑘𝑚h− 1

After how many seconds did the car reach its maximum speed? What was it?1.5 𝑠𝑒𝑐𝑜𝑛𝑑𝑠89𝑘𝑚h− 1

Answers - Easy

Answers - Hard 𝑃𝑙𝑒𝑎𝑠𝑒𝑡𝑟𝑦 h𝑡 𝑒𝑎𝑐𝑡𝑖𝑣𝑜𝑡𝑒!

Graphing Real World Simultaneous EquationsUsually problems with simultaneous equations discuss their intersection. This could imply when a person catches up with another, an object lands on a slope or another meaning.ExampleA boy on a bicycle cycles down a slope and leaves the same time as his dog. The dog runs at and the boy on his bicycle has travelled metres. Draw a graph representing this and state the point in time when the boy passes the dog.

We should write the dogs’ statement in terms of how far it has travelled.

𝑦=4 𝑥

Graphing Real World Simultaneous EquationsNow we have both statements in terms of distance, and (where is the number of metres). Let’s use the formulae this time to find the vertex for .

h=− 𝑏2𝑎,𝑘=4 𝑎𝑐−𝑏

2

4𝑎

h=¿− 𝑏2𝑎

=¿−

2

(2∙ 12 )=¿¿

−2

𝑘= 4𝑎𝑐−𝑏2

4 𝑎¿ 0−2

2

4 ∙12¿−2

Vertex:

Graphing Real World Simultaneous EquationsLet’s draw the lines.

𝑦=4 𝑥𝑦= 𝑥2

2+2𝑥

𝑉𝑒𝑟𝑡𝑒𝑥 :(−2 ,−2)Oh no, the vertex is off the graph! But it’s okay, because we know , so the next point is…

222

62

10

(0 ,0)We know the boy catches up with his dog at the intersection, so it takes…

4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Answers - Easy

Answers - Hard