Equation of a Line Thm. 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)

Ex. Graph: b)

Ex. Graph: c)

Slope Thm. The slope between (x 1,y 1 ) and (x 2,y 2 ) is Ex. Find the slope between (-2,0) and (3,1).

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

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 (x 1,y 1 ) has the equation y – y 1 = m(x – x 1 )  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, and b)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?

Ex. An appliance company determines that the total cost, in dollars, of producing x blenders is C = 25x Explain 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 variable y is called the dependent variable

Ex. Determine whether the relation is a function: a) b) y 3230x

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) x 2 + y = 1 f) x + y 2 = 1

Rather than writing y = 7x + 2 we can express a function as f (x) = 7x + 2  This is called Function Notation

Ex. Let g(x) = -x 2 + 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).

Finding the domain means determining all possible x’s that can be put into the function Ex. Find the domain of the function f : {(-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:

Ex. Find the domain of the function c)

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) = x 2 + 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) = x 2 – 4x + 7, find

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 function a) b)

Def. The zeroes of a function f are the x- values for which f (x) = 0. Ex. Find the zeroes of the function a) f (x) = 3x 2 + x – 10

Ex. Find the zeroes of the function b) c) This is where the graph crosses the x-axis.

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

Ex. Use a calculator to approximate the relative minimum of the function f (x) = 3x 2 – 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. (x1,y1)(x1,y1) (x2,y2)(x2,y2)

The points can be connected using a secant line (x1,y1)(x1,y1) (x2,y2)(x2,y2)

The average rate of change is the slope between the points (x1,y1)(x1,y1) (x2,y2)(x2,y2)

Ex. Find the average rate of change of f (x) = x 3 – 3x from x 1 = -2 to x 2 = 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 t 1 = 4 to t 2 = 9.

Def. A function f (x) is even if f (-x) = f (x).  The graph will have y-axis symmetry

Def. A function f (x) is odd if f (-x) = - f (x).  The graph will have origin symmetry

Ex. Determine if the function is even, odd, or neither: a) g(x) = 3x 3 – 2x b) h(x) = x 2 + 1

Ex. Determine if the function is even, odd, or neither: c) f (x) = x 3 – 4x + 8

Practice Problems Section 2.3 Problems 3, 10, 15, 33, 54, 63, 71, 89, 93

Parents Functions We 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

The identify function is f (x) = x

The squaring function is f (x) = x 2  The domain is all real numbers  The range is all nonnegative numbers  The graph is even and has y-axis symmetry

The cubic function is f (x) = x 3  The domain is all real numbers  The range is all real numbers  The graph is odd and has origin symmetry

The reciprocal function is  The domain is all nonzero numbers  The range is all nonzero numbers  The graph is odd and has origin symmetry

Ex. Sketch a graph of

Def. The greatest integer function,, is defined as

The graph looks like this: This type of function is called a step function

Practice Problems Section 2.4 Problems 29, 43