Chapter 1 – Quadratics

The Questions in this revision are taken from the book so you will be able to find the answers in there.

This should be used in conjunction with your formula booklet!

Chapter 1 Quadratics Part A

A quadratic is of the form y = ax2 +bx + c

When solving a quadratic you are trying to find – the roots, a solution, the x-intercepts. These are the terms that you may come across. Each time y = 0.

e.g. Solve for x: 9x2 – 12x + 4 = 0

Solve for x: (x + 1)2 = 2x2 -5x + 11 Hint: get the equation in the form as above.

You need to be able to solve a quadratic by been able to complete the square which means the quadratic is in the form a(x - h)2 + k = 0.A quadratic in the completed square form can be solved by getting x by itself.

e.g. 3x2 = 6x + 4

You need to be able to solve a quadratic using the quadratic formula which

is

e.g. Hint: get the equation in the form ax2 +bx + c = 0.

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Chapter 1 Quadratics Part B

The Discriminant Δ = b2 – 4acIf Δ > 0 then there are two real roots i.e. the graph crosses the x-axis at two points.

If Δ = 0 then there is only one real root, i.e. the graph touches the x-axis only once.

If Δ < 0 then there are no roots, i.e. the graph does not cross or touch the x-axis.

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Chapter 1 Quadratics Part C

Quadratic function.

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Also when

finding the y-intercept

x = ?

When finding the

x-intercepts y = ?

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Chapter 1 Quadratics Part C

Quadratic function - graphing.

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Also write the equation of the axis of symmetry

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Chapter 1 Quadratics Part D

Finding the Quadratic Equation From the Graph.

If you know the the x & y intercepts you can use this to find the equation of the graph.

Or if you know the vertex with or without the x or y intercepts you can use these to find the equation of the graph.

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Chapter 1 Quadratics Part E

Where Graphs Meet – Find the Intersection of two FunctionsUsing Technology or Simultaneous Equations

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Chapter 1 Quadratics Part F

Problem Solving With Quadratics.

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Chapter 1 Quadratics Part G

Quadratic Optimization.Finding the maximum or minimum value of a quadratic is optimisation v

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For further Questions do Review Exs 1B & 1C.

Chapter 2 - Functions

This should be used in conjunction with your formula booklet!

Chapter 2 Functions Part A Relations & Functions

A Relation is any set of points that connect two variables. The variables are often generated by an equation connecting the variables x & y. In this case the relation is a set of points (x,y) in the Cartesian Plane

A Function, is sometimes called a Mapping, is a relation in which every x-value only has one y-value.

Function Test – If a vertical line only goes through the graph once then it is a function

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Part B Function Notation

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Chapter 2 FunctionsPart C Domain & Range.Domain is all the values that x can take for a given equation.Range is all the values y can take for a given equation.

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See examples 6 & 7

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Chapter 2 Functions Part E Sign Diagrams.Sign Diagrams show us what the graph does i.e. is does it above your below the x-axis.You have to find from the equation the x-intercepts and anywhere where the graph is undefined. Then on either side of these points choose a value for x and find the y value for that point. If positive place a “+” at that point on the sign diagram; if y is negative place a “-” on the sign diagram.

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Dashed line means the graph is undefined at this point

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Chapter 2 Functions Part D Composite Functions.

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Chapter 2 Functions Part F Rational Functions.Dividing a linear function by a linear function gives a Rational Function. e.g.

Rational Functions are characterised by vertical and horizontal asymptotes that the graph gets closer and closer to but never reaches.

Vertical Asymptote – Using the above general equation, this is found when the denominator = 0. So for the above cx + d = 0 so x = -d/c, is the equation of the vertical asymptote.

Horizontal asymptote – Using the above general equation, the equation of this asymptote is y = a/c.

To find the x-intercepts, use y = 0.To find the y-intercepts, let x = 0.

Understand how the asymptotesand the intercepts were found forthe graph to the right.

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Chapter 2 Functions Part F

Note: for the vertical and horizontal asymptotes are the y &

x-axis respectively.

For the vertical asymptote is x = c and the

horizontal asymptote is y = d. (These ideas are also in Chp 5 Transformations.

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Chapter 2 Functions Part G Inverse Functions.

When you find an inverse function and it is the same as the original Function then this is called a self-inverse function.

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Then solve for y

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vFor further Questions do Review Exs 2B & 2C.