Lecture 6 sections 2.3 and 2.7 graphs of lines and transformations

MATH 108

Section 2.3

Lines

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SLOPE–INTERCEPT FORM OF THEEQUATION OF A LINE

The slope-intercept form of the equation of the line with slope m and y-intercept b is

y mx b .


EXAMPLE 5

Graph

Graphing by Using the Slope andy-intercept

22

3y x

Solution


HORIZONTAL AND VERTICAL LINES

An equation of a horizontal line through (h, k) is y = k.

An equation of a vertical line through (h, k) is x = h.

Graph the equation: x = - 2

Graph the equation: x = 0

Graph the equation: y = 5

Graph the equation: y = 0


SLOPE & DIRECTION

1. Scanning graphs from left to right, lines with positive slopes rise and lines with negative slopes fall.

2. The greater the absolute value of the slope, the steeper the line.

3. The slope of a vertical line is undefined.

4. The slope of a horizontal line is 0.

Find the equation of a line with slope -3 and containing the point (-1, 4).

Find an equation for the line that contains the point ( 1,3) and is

parallel to the lin 3 4 1e .2x y

Find an equation for the line that contains the point ( 1,3) and is

perpendicular to the li 2ne 3 1 .4x y

1 1y y m x x

Section 2.7

Transformations of Functions


EXAMPLE 1 Graphing Vertical Shifts

Let , 2, and 3.f x x g x x h x x Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.


EXAMPLE 1 Graphing Vertical Shifts

Solution continued

Graph the equations.The graph of y = |x| + 2 is the graph of y = |x| shifted two units up. The graph of y = |x| – 3 is the graph of y = |x|shifted three units down.


VERTICAL SHIFT

Let d > 0. The graph of y = f (x) + d is the graph of y = f (x) shifted d units up, and the graph of y = f (x) – d is the graph of y = f (x) shifted d units down.


EXAMPLE 2 Writing Functions for Horizontal Shifts

Let f (x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2.

A table of values for f, g, and h is given on the next slide. The graphs of the three functions f, g, and h are shown on the following slide.

Describe how the graphs of g and h relate to the graph of f.


EXAMPLE 2 Writing Functions for Horizontal Shifts


HORIZONTAL SHIFT

The graph of y = f (x – c) is the graph of y = f (x) shifted •c units to the right, if c > 0 •c to the left, if c < 0.


EXAMPLE 3

Sketch the graph of the function

2 3.g x x

Solution

Step 1 Identify and graph the known function Choose .f x x

Graphing Combined Vertical and Horizontal Shifts


EXAMPLE 3

Step 2 Identify the constants d and c in the transformation g (x) = f (x – c ) + d.

Step 3 Since c = –2 < 0, the graph ofis the graph of f shifted horizontally two units to the left.


Solution continued

d


EXAMPLE 3

Solution continued


Step 4 Because d = –3 < 0, graph

by shifting the graph of

three units down. 2 3g x x


REFLECTION IN THE x-AXIS


REFLECTION IN THE y-AXIS


EXAMPLE 5 Stretching or Compressing a Function Vertically

Solution

Sketch the graphs of f, g, and h on the same coordinate plane, and describe how the graphs of g and h are related to the graph of f.

Let


EXAMPLE 5 Stretching or Compressing a Function Vertically

Solution continued


VERTICAL STRETCHING OR COMPRESSING

The graph of y = a f (x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is

1. A vertical stretch away from the x-axis if a > 1;

2. A vertical compression toward the x-axis if 0 < a < 1.

If a < 0, the graph of f is first reflected in the x-axis, then vertically stretched or compressed.


HORIZONTAL STRETCHING OR COMPRESSING

The graph of y = f (bx) is obtained from thegraph of y = f (x) by multiplying the x-coordinate

of each point on the the graph of y = f (x) by and leaving the y-coordinate unchanged. The result is

1. A horizontal stretch away from the y-axis if 0 < b < 1;

2. A horizontal compression toward the y-axis if b > 1.

1

b


EXAMPLE 7 Combining Transformations

Sketch the graph of the functionf (x) = 3 – 2(x – 1)2.

Solution

Step 1 y = x2

Identify a related function.Step 2 y = (x – 1)2

Shift right 1.



Solution continued

Step 3 y = 2(x – 1)2

Stretch vertically by a factor of 2.

Step 4 y = –2(x – 1)2

Reflect in x-axis.



Solution continued

Step 5 y = 3 – 2(x – 1)2 Shift three units up.

Technology

Lecture 6 sections 2.3 and 2.7 graphs of lines and transformations