Practice Problems

Tangent = because the line has the same slope as the curve at that one point.
Secant = because the slope of this line is found using 2 points on the curve.

Grey Line = tangent line
Green Line = Secant

The average rate of change can be found by means of the slope formula.

$ slope = \frac{ \Delta y }{ \Delta x} = \frac{3-6}{5--1}= \frac{-3}{6} = -\frac{1}{2} $

Slope: $ slope = \frac{3-1}{9-1}= \frac{2}{8} = \frac{1}{4} $

Equation in point slope form: $ (y-3) = \frac{1}{4} (x -9) $

Alternate equation in point slope form: $ (y-1) = \frac{1}{4} (x -1) $

Equation in slope intercept form: $ y= \frac{1}{4} +\frac{3}{4} $

• A) Sketch the graph of $$y= (x-1)^2 +2 $$

• B) Plot the point x = -1 and label it "a"
• C) Plot the point x= 1 and label it "b"
• D) Find the equation of the secant line between points "a" and points "b"

Answers for B,C, and D

Slope: $ $slope = \frac{2-4}{1--1}= \frac{-2}{2} = -1$$
Equation: (y-2)=-1(x-1) or y=-x+3

Graph Secant Line

• x =1
• x = 0
• x = -2
• x = 2

Points	Slope
Choice a (1,1) Given point (-1,-1)	$\frac{-1-1}{-1-1}= \frac{-2}{-2} = 1$
Choice b (0,0) Given point (-1,-1)	$\frac{-1-0}{-1-0}= \frac{-1}{-1} = 1$
Choice c (-2,8) Given point (-1,-1)	$\frac{-1-8}{-1--2}= \frac{-9}{1} = -9$
Choice c (2,8) Given point (-1,-1)	$\frac{-1-8}{-1-2}= \frac{-9}{-3} = 3$ This secant's slope is the greatest and therefore is the answer