Unit 2 - Functions
Required Prior Knowledge
Questions
If \(y=3x^{2}-2x+1\), find the value of \(y\) when
a) \(x=0 \)
b) \(x=3\)
c) \(x=-2\)
Determine if \(\left(2,16\right)\) satisfies the quadratic function \(y=3x^{2}+2x\).
If \(y=x^{2}-2x+3\), find the value(s) of \(x\) when \(y=2\).
Factorise \(2x^{2}-x-6\).
Write \(3x^{2}-12x-29\) in the form \(a\left(x-h\right)^{2}+k\).
Solutions
Get Ready
Questions
What different types of functions do you already know?
Write down as much information as you can about each of them.
Solutions
Notes
Quadratic functions are any function that can be written in the form \(f\left(x\right)=ax^{2}+bx+c\), where \(a\ne 0\).
This is known as the general or standard form of a quadratic.
All quadratics have the same general shape, known as a parabola.
If \(a>0\) then the quadratic is concave up and will have this shape with a local minimum.
Otherwise, if \(a<0\) the quadratic is concave down and will have this shape with a local maximum.
The natural domain of all quadratic functions is \(\left\{x:x\in\mathbb{R}\right\}\).
The range of a quadratic depends on its concavity:
A concave up quadratic with minimum at \(\left(h,k\right)\) has range \(\left\{y:y\ge k, y\in\mathbb{R}\right\}\)
A concave down quadratic with maximum at \(\left(h,k\right)\) has range \(\left\{y:y\le k, y\in\mathbb{R}\right\}\)
All quadratics have a line of symmetry which passes through the vertex (local minimum or local maximum).
For a summary of the different forms of a quadratic, watch the video below.
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The general form of a quadratic function is \(y=ax^{2}+bx+c\) where \(a\ne 0\) and \(b,c\in\mathbb{R}\).
The \(y\)-intercept is \(\left(0,c\right)\).
The axis of symmetry is \(x=-\frac{b}{2a}\).
The \(x\)-coordinate of the vertex is \(-\frac{b}{2a}\).
We can find the roots by factorising or using the quadratic formula.
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The Root Form of a quadratic function has the form \(y=a\left(x-p\right)\left(x-q\right)\) where \(a\ne 0\) and \(p,q\in\mathbb{R}\).
The roots of the quadratic are at \(\left(p,0\right)\) and \(\left(q,0\right)\).
The axis of symmetry is through the midpoint of the roots at \(x=\frac{p+q}{2}\).
The \(x\)-coordinate of the vertex is \(\frac{p+q}{2}\).
The \(y\)-intercept is \(\left(0,apq\right)\).
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The Vertex Form of a quadratic function has the form \(y=a\left(x-h\right)^{2}+k\) where \(a\ne 0\) and \(h,k\in\mathbb{R}\).
The vertex (either maximum or minimum) of the quadratic is at \(\left(h,k\right)\).
The axis of symmetry is through the vertex at \(x=h\).
The \(y\)-intercept is \(\left(0,ah^{2}+k\right)\).
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A special case of a quadratic function has the form \(y=a\left(x-h\right)^{2}\) where \(a\ne 0\) and \(h\in\mathbb{R}\).
This is both Root Form and Vertex Form.
There is a single root which is the vertex at \(\left(h,0\right)\).
Fill In The Blanks
Fill in the missing parts of the table below.
| \(x^2 + bx + c\) | \((x+d)(x+e)\) | \((x+p)^2 + q\) | \(x\)-int | \(y\)-int | Axis of Sym. | Turning Point | Sketch |
|---|---|---|---|---|---|---|---|
| \(x^2 - 6x + 5\) | \((x-1)(x-5)\) ? |
\((x-3)^2 - 4\) ? |
\((1,0), (5,0)\) ? |
\((0,5)\) ? |
\(x=3\) ? |
\((3, -4)\) ? |
?
|
| \(x^2 + 4x + 3\) ? |
\((x+1)(x+3)\) | \((x+2)^2 - 1\) ? |
\((-1,0), (-3,0)\) ? |
\((0,3)\) ? |
\(x=-2\) ? |
\((-2, -1)\) ? |
?
|
| \(x^2 + 6x + 8\) ? |
\((x+2)(x+4)\) ? |
\((x+3)^2 - 1\) | \((-2,0), (-4,0)\) ? |
\((0,8)\) ? |
\(x=-3\) ? |
\((-3, -1)\) ? |
?
|
| \(x^2 - 9\) ? |
\((x-3)(x+3)\) ? |
\(x^2 - 9\) ? |
\((3,0), (-3,0)\) | \((0,-9)\) ? |
\(x=0\) ? |
\((0, -9)\) ? |
?
|
| \(x^2 - 4x\) ? |
\(x(x-4)\) ? |
\((x-2)^2 - 4\) ? |
\((0,0), (4,0)\) ? |
\((0,0)\) ? |
\(x=2\) ? |
\((2, -4)\) ? |
|
| \(x^2 - 4x + 4\) | \((x-2)^2\) ? |
\((x-2)^2\) ? |
\((2,0)\) ? |
\((0,4)\) ? |
\(x=2\) ? |
\((2, 0)\) ? |
?
|
| \(x^2 + 2x - 8\) ? |
\((x+4)(x-2)\) ? |
\((x+1)^2 - 9\) ? |
\((-4,0), (2,0)\) ? |
\((0,-8)\) ? |
\(x=-1\) ? |
\((-1, -9)\) |
?
|
| \(x^2 + 8x + 12\) ? |
\((x+2)(x+6)\) | \((x+4)^2 - 4\) ? |
\((-2,0), (-6,0)\) ? |
\((0,12)\) ? |
\(x=-4\) ? |
\((-4, -4)\) ? |
?
|
Examples and Your Turns
Example
Sketch the graph of the function$$f\left(x\right)=2\left(x-3\right)\left(x+1\right)$$without using your GDC.
Your Turn
Sketch the graph of the function$$f\left(x\right)=-\left(x+2\right)\left(x+5\right)$$without using your GDC.
Example
Sketch the graph of the function$$f\left(x\right)=-3\left(x-2\right)^{2}-4$$without using your GDC.
Your Turn
Sketch the graph of the function$$f\left(x\right)=\left(x+1\right)^{2}+2$$without using your GDC.
Example
Find the domain and range of the function $$y=-2x^{2}+4x-3$$
Your Turn
Find the domain and range of the functions
\(y=x^{2}-4x+2\)
\(y=-\left(x+2\right)^{2}-3\)
\(y=-3x^{2}+6x-1\)
Example
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Your Turn
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Example
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Your Turn
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Example
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Your Turn
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Your Turn
Find the equation of the quadratic whose graph cuts the \(x\)-axis at \(4\) and \(-3\), and which passes through the point \(\left(2,-20\right)\). Give your answer in the form \(y=ax^{2}+bx+c\).
Your Turn
Find the equation of the quadratic function with \(x\)-intercepts at \(\left(-4,0\right)\) and \(\left(6,0\right)\), and which passes through the point \(\left(-1,-8.75\right)\).
Your Turn
Find the quadratic function with vertex \(\left(2,11\right)\) and passes through the point \(\left(-1,-7\right)\).
Your Turn
Find the quadratic function with vertex \(\left(-1,-9\right)\) and passes through the point \(\left(3,7\right)\).
Example
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Your Turn
Find the equation of the quadratic function with the following graph. Give your answer in the form \(f\left(x\right)=ax^{2}+bx+c\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why does the sign of \(a\) determine whether the parabola is concave up or concave down?
Explain why the axis of symmetry is located at the midpoint of the roots (if they exist).
If a quadratic function has no real roots, is it possible to determine the equation of the function from the graph? What information would you need?
Explain is the general process for finding the equation of a quadratic function when given the graph.
Common Mistakes / Misconceptions
Sign errors in vertex and root form. Remember that if \(y=\left(x+3\right)^{2}-2\) then the vertex is at \(\left(-3,-2\right)\) where the sign of the \(x\) coordinate changes but the sign of the \(y\) coordinate does not.
Connecting This to Other Skills
Quadratic functions link to many skills from across the course such as:
Solving Quadratic Equations (prior knowledge)
Skill 2.14 on solving more quadratic equation, with complex roots
Skill 2.15 Solving Quadratic Inequalities
Skill 2.16 the Discriminant
Skill 2.18 Function Transformations
Skill 4.4 Differentiating polynomials
Skill 4.14 Stationary Points
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?