# Linear and Quadratic Functions

## Linear Functions

A linear function is a function that can be written in the
form f (x) = mx + b .

Its graph is a straight line.

m = slope

b = y-intercept

Recall: If a line passes through (x_{1} , y_{1} ) and
(x_{2} , y_{2} ) , its slope m is given by

The sign of the slope indicates whether the line is an
increasing or a decreasing function.

m < 0

m > 0

Also recall:

1. The slope of any horizontal line is 0.

2. The slope of any vertical line is undefined.

**Example 1: **Graph the equation using its slope and
intercept:

** Two useful forms for the equation of a line :**

1. y = mx + b (slope- intercept form )

2. y - y_{1} = m(x - x_{1} ) (point-slope form)

**Note:** We are working with functions, so we want y
on the left-hand side by itself.

This means we need to rearrange our equations into slope - intercept form so we
can write

f (x) = mx + b .

**Parallel Lines
**Two lines are parallel if they never cross. In terms of equations, two lines
are parallel if

they have __the same_____ slope.

**Perpendicular Lines
**Two lines are perpendicular if they meet at a right angle. In terms of
equations, two lines

with slopes m

_{1}and m

_{2}are perpendicular if and only if

m

_{1}* m

_{2}= -1 .

**Example 2:** Find the linear function f such that f
(1) = 5 and f (2) = 8 . What is the inverse of this

function?

**Example 3:**Write the equation of the linear function
f(x) that passes through the point (-4,5)

and is parallel to the line 4x + 3y = 7 .

**Example 4:** Find the linear function f that is
perpendicular to the line containing (4,-2) and

(10,4) and passes through the midpoint of the line segment connecting these
points.

## Quadratic Functions

A quadratic function is a function which can be written in
the form f (x) = ax^{2} + bx + c

( a ≠ 0 ). Its graph is a parabola .

Every quadratic function also know as a parabola is
written as f (x) = ax^{2} + bx + c or can be written in

standard form: f (x) = a(x - h)^{2} + k . The vertex is (h, k) . The y-intercept
is f(0). The axis of symmetry is

the equation x = h.

^{2}+ bx + c , or f (x) = a(x - h)

^{2}+ k :

· The graph opens up if a > 0 .

· If |a| is larger, is the parabola is narrower ; if |a| is smaller, is the parabola is wider.

a is referred to as the shape factor .

Shortcut:

For f (x) = ax

^{2}+ bx + c , the vertex is ,. So the axis of symmetry is .

You should be able to identify the following :

· Direction the graphs opens(upwards or downwards)

· Whether the function has a maximum or a minimum

· y-intercept

· coordinates of the vertex

· equations of the axis of symmetry

· maximum or minimum

**Example 5:** Sketch the graph of f (x) = 3x^{2} + 6x +
7 by finding the six features of the function.

**Example 6:** Sketch the graph of f (x) = -2x^{2} + 12x
- 16 by finding the six features of the function.

**Example 7:** Find the quadratic function such that
the axis of symmetry is x = - 2 , the y-intercept is - 6

and there is only one x-intercept.

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