equation of a line thm. a line has the equation y = mx + b, where m = slope and b = y-intercept. ...

Equation of a LineThm. A line has the equation y = mx + b,

where m = slope and b = y-intercept.

This is called Slope-Intercept Form

Ex. Find the slope and y-intercept:a) y = 3x + 1

b) 2x + 3y = 1

Ex. Graph:a) 2 1y x

4

2

-2

-4

-5 5

Ex. Graph:b) 2y

4

2

-2

-4

-5 5

Ex. Graph:c) 2x y

4

2

-2

-4

-5 5

SlopeThm. The slope between (x1,y1) and (x2,y2) is

Ex. Find the slope between (-2,0) and (3,1).

2 1

2 1

y ym

x x

• Rising line has positive slope:

• Falling line has negative slope:

• Horizontal line as slope of 0:

• Vertical line has undefined slope:

A vertical line has an equation like x = 3, and can’t be written as y = mx + b

25 , 4

378,

0 05 3,

820 0,

Thm. Parallel lines have the same slope.

Thm. Perpendicular lines have slopes that are negative reciprocals.

Ex. Are the lines parallel, perpendicular, or neither? (3,-1) to (-3,1) and (0,3) to (-1,0)

Thm. A line with m = slope that passes through the point (x1,y1) has the equation

y – y1 = m(x – x1)

This is called Point-Slope Form

Ex. Write the equation in Slope-Intercept Form:a) slope = 3, contains (1,-2)

Ex. Write the equation in Slope-Intercept Form:b) contains (2,5) and (4,-1)

Ex. Find equation of the lines that pass through (2,-1) are:a) parallel to, andb) perpendicular to, the line 2x – 3y = 5

Ex. The maximum slope of a wheelchair ramp is . A business is installing a ramp that rises 22 in. over a horizontal distance of 24 ft. Is the ramp steeper than required?

112

Ex. An appliance company determines that the total cost, in dollars, of producing x blenders is

C = 25x + 3500Explain the significance of the slope and y-intercept.

Ex. A college purchases exercise equipment worth $12,000. After 8 years, the equipment is determined to have a worth of $2000. Express this relationship as a linear equation. How many years will pass before the equipment is worthless?

Practice Problems

Section 2.1

Problems 9, 21, 41, 51, 69, 107, 109

Functions

Def. (formal) A function f from set A to set B is a relation that assigns to each element x in set A exactly on element y in set B.

Def. (informal) A set of ordered pairs is a function if no two points have the same x-coordinate

Set A(the x’s)

The input

Domain

Set B(the y’s)

The output

Range

x is called the independent variabley is called the dependent variable

Ex. Determine whether the relation is a function:

a)

b)

40-375y

3-1230x

2

-2

Ex. Determine whether the relation is a function:

c) x is the number of representatives from a state, y is the number of senators from the same state

d) x is the time spent at a parking meter, y is the cost to park

Ex. Determine whether the relation is a function:

e) x2 + y = 1

f) x + y2 = 1

Rather than writingy = 7x + 2

we can express a function asf (x) = 7x + 2

This is called Function Notation

Ex. Let g(x) = -x2 + 4x + 1, find

a) g(2)

b) g(t)

c) g(x + 2)

The next example is called a piecewise function because the equation depends on what we are plugging in.

Ex. Let , find f (-1),

f (0), and f (1).

2 1 0

1 0

x xf x

x x

Finding the domain means determining all possible x’s that can be put into the function

Ex. Find the domain of the functionf : {(-3,0), (-1,4), (0,2), (2,2), (4,-1)}

Often, finding the domain means finding the x’s that can’t be used in the function

Ex. Find the domain of the function

a)

b) Volume of a sphere:

1

5g x

x

343V r

Ex. Find the domain of the functionc) 24h x x

Ex. You’re making a can with a height that is 4 times as long as the radius. Express the volume of the can as a function of height.

Ex. When a baseball is hit, the height of a baseball is given by the function f (x) = -0.0032x2 + x + 3, where x is distance travelled (in ft) and f (x) is height (in ft). Will the baseball clear a 10-foot fence that is 300 ft from home plate?

Ex. For f (x) = x2 – 4x + 7, find f x h f x

h

Practice Problems

Section 2.2

Problems 9, 15, 29, 35, 59, 79, 87, 93

Graph of a Function

Ex. Using the graph, find:

a) domain

b) range

c) f (-1), f (1), and f (2)

The graph of a function will pass the vertical line test – all vertical lines will pass through the graph at most once.

Ex. Determine if this is the graph of a functiona)

4

2

b) 4

2

-5

Def. The zeroes of a function f are the x-values for which f (x) = 0.

Ex. Find the zeroes of the functiona) f (x) = 3x2 + x – 10

Ex. Find the zeroes of the functionb)

c)

This is where the graph crosses the x-axis.

2 3

5

th t

t

210g x x

Let discuss increasing, decreasing, relative minimum, and relative maximum

4

3

2

1

2 4 6 8 10

Ex. Use a calculator to approximate the relative minimum of the function f (x) = 3x2 – 4x – 2.

Earlier, we worked with slope as the rate of change of a line

If the graph is nonlinear, we still want to talk about rate of change, but this slope is different at every point.

We can discuss the average rate of change between two points.

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

-0.2

0.5 1 1.5 2 2.5 3 3.5 4

(x1,y1)

(x2,y2)

The points can be connected using a secant line

(x1,y1)

(x2,y2)

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

-0.2

0.5 1 1.5 2 2.5 3 3.5 4

The average rate of change is the slope between the points

(x1,y1)

(x2,y2)

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

-0.2

0.5 1 1.5 2 2.5 3 3.5 4

2 1

2 1

y ym

x x

Ex. Find the average rate of change of f (x) = x3 – 3x from x1 = -2 to x2 = 0.

Ex. The distance s (in feet) a moving car has traveled is given by the function , where t is time (in seconds). Find the average speed from t1 = 4 to t2 = 9.

3220s t t

Def. A function f (x) is even if f (-x) = f (x).

The graph will have y-axis symmetry

4 22y x x

1.5

1

0.5

-0.5

-1

-1 1

Def. A function f (x) is odd if f (-x) = - f (x).

The graph will have origin symmetry1.5

1

0.5

-0.5

-1

-1 1

3 2y x x

Ex. Determine if the function is even, odd, or neither:

a) g(x) = 3x3 – 2x

b) h(x) = x2 + 1

Ex. Determine if the function is even, odd, or neither:

c) f (x) = x3 – 4x + 8

Practice Problems

Section 2.3

Problems 3, 10, 15, 33, 54, 63, 71, 89, 93

Parents FunctionsWe are going to talk about some basic

functions, and next class we will expand upon them.

Earlier, we saw that a function f (x) = ax + b is linear

The domain of a linear function is all real numbers, and the range is all real numbers

The constant function is f (x) = c

The graph is a horizontal line

4

2

-2

The identify function is f (x) = x

2

-2

The squaring function is f (x) = x2

The domain is all real numbersThe range is all nonnegative numbersThe graph is even and has y-axis symmetry

4

2

The cubic function is f (x) = x3

The domain is all real numbersThe range is all real numbersThe graph is odd and has origin symmetry

2

-2

The reciprocal function is

The domain is all nonzero numbersThe range is all nonzero numbersThe graph is odd and has origin symmetry

1f x

x

3

2

1

-1

-2

-2 2

Ex. Sketch a graph of 2 3 1

4 1

x xf x

x x

6

4

2

-2

-4

-6

-5 5

Def. The greatest integer function, , is defined as

x

the greatest integer

less than or equal tof x x

x

1.5

85 5 3.7

The graph looks like this:

This type of function is called a step function

Practice Problems

Section 2.4

Problems 29, 43