Activity 2 - 4

1
Activity 2 - 4

• Family of Functions

2
5-Minute Check on Activity 2-3

1. What is the formula for slope-intercept form of a
   line?
2. How do you find the y-intercept of a line?
3. How do you find the x-intercept of a line?
4. How can we use our calculator to find the
   intercepts?

? y y2
y1 Slope m ---------- --------------
y mx b
? x x2 x1
Plug x 0 into the equation and solve for y
Plug y 0 into the equation and solve for x
y-intercept look for the x0 value in the table
(2nd graph) X-intercept use 2nd trace
(calculate) option 2 zeros to let the calculator
estimate an x-intercept
Click the mouse button or press the Space Bar to
display the answers.
3
Objectives

• Identify the effect of changes in the equation of
  a line on its graph
• Identify the effect of changes in the graph of a
  line on its equation
• Identify the change in the graph and the equation
  of a basic function as a translation, reflection
  or vertical stretch or shrink

4
Vocabulary

• Vertical Shift a constant is added (shift up)
  or subtracted (shift down) to each output value
• Horizontal Shift a constant is added (shift
  left) or subtracted (shift right) to each input
  value
• Reflection a flip across an axis algebraically
  a reflection across the x-axis occurs if y f(x)
  f(-x)
• Stretch Factor is called a when the graph of y
  f(x) changes to y a?f(x)
• Vertical Stretch when the graph of y f(x)
  changes to y a?f(x) and a gt 1
• Vertical Shrink when the graph of y f(x)
  changes to y a?f(x) and 0 lt a lt 1
• Transformations any translations (horizontal or
  vertical shifts), reflections and vertical
  stretches or shrinks

5
Activity

• A primary objective of this textbook is to help
  you develop a familiarity with the graphs,
  equations, and properties of a variety of
  functions, including linear, quadratic,
  exponential, and logarithmic. You will group
  these functions into families and identify the
  similarities within a family and the differences
  between families.
• We will continue to explore the family of linear
  functions.

6
Vertical Shifts Revisited

• Given y f(x) 2x
• Graph the function
• Determine the slope and intercepts
• Graph Y2 2x 3 and Y3 2x 4
• Compare the graphs (slope and intercepts)

m 2 y-intercept 0
x-intercept 0
m 2 y-intercept -3
x-intercept 3/2
m 2 y-intercept 4
x-intercept -2
7
Horizontal Shifts Revisited

• Given y f(x) 2x
• Graph the function
• Determine the slope and intercepts
• Graph Y2 2(x 3) and Y3 2(x 3)
• Compare the graphs (slope and intercepts)

m 2 y-intercept 0
x-intercept 0
m 2 y-intercept -6
x-intercept 3
m 2 y-intercept 6
x-intercept -3
8
Both Shifts

• Graph each of the following functions in the same
  window. Y1 x2 Y2 x2 6
  Y3 (x 3)2
• How do the graphs compare? Which is shifted
  horizontally?
• What direction?
• Which is shifted vertically?
• What direction?

Y3 (x 3)2
left
Y2 x2 6
up
9
Reflections Across the X-Axis

• The graph of y -x is a reflection of the graph
  of y x across the x-axis
• In general, if the graph of y f(x) is reflected
  across the x-axis, then the equation of the
  resulting graph is y -f(x)
• The reflection is keeping the x-value the same
  and multiplying the output value, y, by negative
  one.

10
X-Axis Reflections

• Given y f(x) 3x 6
• Graph the function
• Determine the slope and intercepts
• Reflect the graph across the x-axis
• Write the equation of the reflection
• Determine the slope and intercepts

m 3 y-intercept 6
x-intercept -2
y -3x - 6
m -3 y-intercept -6
x-intercept -2
11
Reflections Across the Y-Axis

• The graph of y -x is also a reflection of the
  graph of y x across the y-axis
• In general, if the graph of y f(x) is reflected
  across the y-axis, then the equation of the
  resulting graph is y f(-x)
• The reflection is keeping the y-value the same
  and multiplying the input value, x, by negative
  one.

12
Y-Axis Reflections

• Given y f(x) 3x 6
• Graph the function
• Determine the slope and intercepts
• Reflect the graph across the y-axis find f(-x)
• Write the equation of the reflection
• Determine the slope and intercepts

m 3 y-intercept 6
x-intercept -2
y -3x 6
m -3 y-intercept 6
x-intercept 2
13
Vertical Stretches and Shrinks

• A graph is stretched vertically when the function
  (output value) is multiplied by a constant, a gt 1
• A graph is shrunk vertically when the function
  (output value) is multiplied by a constant, 0 lt a
  lt 1
• A graph is flipped and stretched vertically when
  the function (output value) is multiplied by a
  constant, a lt -1
• A graph is flipped and shrunk vertically when the
  function (output value) is multiplied by a
  constant, -1 lt a lt 0

14
Vertical Stretches

• Given y f(x) x
• Graph the function
• Determine the slope and intercepts
• Graph Y2 2x and Y3 5x
• Compare the three graphs (slope and intercepts)

m 1 y-intercept 0
x-intercept 0
m 2 y-intercept 0
x-intercept 0
m 5 y-intercept 0
x-intercept 0
15
Transformations

• Given y f(x) x
• Graph the function
• Graph Y2 x 3
• Graph Y3 2x 3
• Graph Y4 -2x 3

16
Summary and Homework

• Summary
• Vertical shifts output value constant
• Horizontal shifts (input value constant)
• Reflections
• x-axis x-values same, y-values flip sign
• y-axis y-values same, x-values flip sign
• Shifts (also called translations), reflections
  (flips) and vertical stretches and shrinks are
  called Transformations
• Homework
• Pg 215-7 1 - 7