class 051003

Let y = f(x) = 3 x - 1.

What are the two basic points of this function?

• The points are the x = 0 and x = 1 points (0, -1) and (1, 2).
• Sketch these basic points on a graph and using these points sketch a graph of the function y = f(x).
• The graph is a straight line through the two basic points.
• Show what happens to the basic points if this function is vertically stretched by factor 2. 
• The vertical stretch moves every point twice as far from the x axis (doubling every y value), moving the basic points to (0, -2) and (1, 4).
• Based on these points sketch the graph of y = 2 f(x).
• All we need to do is draw a line thru the two points (0, -2) and ( 1, 4).
• What is the expression for the function y = 2 f(x)?
• y = f(x) = 3x - 1, so y = 2 f(x) = 2 ( 3x - 1) = 6x - 2.

Show what happens to the basic points if the function y = 2 f(x) is then vertically shifted +4 units.

Each point is raised 4 units in the y direction, increasing its y coordinate by 4.  Our points (0, -2) and (1, 4) become (0, -2+4) = (0, 2) and (1, 4+4) = (1, 8).

• Based on these points sketch the graph of the function y = 2 f(x) + 4.
• The graph is constructed by drawing the straight line through the two new basic points.
• This has the effect of raising every point of the y = 2 f(x) graph 4 units. 
• We end up with a line parallel to that of y = 2 f(x), but 4 units higher.
• What is the expression for the function y = 2 f(x) + 4?
• Since f(x) = 3x - 1, the expression for y = 2 f(x) + 4 is y = 2 ( 3x - 1) + 4.
• This simplifies to y = 6 x + 2.

If we wanted to horizontally shift the function 10 units?

The function y = 2 f(x-10) + 4 would accomplish this shift.

The function would be y = 2 f(x-10) + 4. 

Since f(x) = 3x - 1,

f(x-10) = 3 ( x-10) + 1 and

2 f(x-10) + 4 would be

2 ( 3(x-10) - 1) + 4 =

2 (3x - 30 - 1) + 4 = 2 ( 3x - 31) + 4 = 6x - 62 + 4 = 6x - 58.

What is f(3)?

f(x) = 3x - 1 so f(3) = 3 * 3 - 1 = 8.

What is the expression for f(a + b)?

f(a+b) = 3(a+b) - 1 = 3a + 3b - 1.

What is the expression for f(x1)?

f(x) = 3x - 1 so f(x1) = 3 x1 - 1.

What is the expression for f(x2)?

f(x) = 3x - 1 so f(x2) = 3 x2 - 1.

What is the expression for f(x2) - f(x1)?

f(x2) = (3 x2 - 1) and f(x1) = (3 x1 - 1) so

f(x2) - f(x1) = (3 x2 - 1) - (3 x1 - 1)

= 3 x2 - 1 - 3 x1 + 1

= 3 x2 - 3 x1 - 1 + 1; since -1 + 1 = 0 this is the same as

= 3 x2 - 3 x1

= 3 ( x2 - x1).

What is the expression for ( f(x2)- f(x1) ) / ( x2 - x1)?

The numerator is just what we saw above.  We would substitute and simplify to get f(x2) - f(x1) = 3 ( x2 - x1).

So the expression

(f(x2) - f(x1) ) / ( x2 - x1)

= 3 ( x2 - x1) / (x2 - x1)

= 3.

Note that since f(x2) is the y value corresponding to x = x2 (let's say that y2 = f(x2) ) and f(x1) is the y value corresponding to x = x1 (we'll say y1 = f(x1) ):

• ( f(x2) - f(x1) )  represents ( y2 - y1 ), the change in y coordinates.
• x2 - x1 is the change in x coordinates so
• ( f(x2) - f(x1) ) / ( x2 - x1) represents (change in y coordinate / change in x coordinate) = rise / run = slope, and is also equal to the average rate of change of y with respect to x.

On a graph of y vs. x, (x1, f(x1) ) and (x2, f(x2) ) are points on the graph, and the slope of the line segment connecting these points is ( f(x2) - f(x1) ) / (x2 - x1).

For the linear function y = 3 x - 1, this expression simplifies to just 3, which proves what we thought we already knew--that between any two points of this graph the slope is 3.