Download STATION 1 For linear transformation from the parent linear function

STATION 1
For linear transformation from the parent linear function f ( x)  x , the general
form representing linear transformation can be described by
𝑔(𝑥 ) = 𝑎 ∙ 𝑓 [𝑏(𝑥 − 𝑐 )] + 𝑑
1. Describe what transformations a controls and give an example
2. Describe what transformations b controls and give an example
3. Describe what transformations c controls and give an example
4. Describe what transformations d controls and give an example
5. Write an example using all 4 transformations and describe the changes
being made to the parent linear function 𝑓(𝑥 ) = 𝑥
STATION 1 ANSWER KEY
1. If a  1 then the line is vertically stretched by a factor of a
If 0  a  1 then the line is vertically compressed by a factor of a
If a  0 then the line is reflected over the x-axis
2. If b  1 then the line is horizontally compressed by a factor of
If 0  b  1 then the line is horizontally stretched by a factor of
If b  0 then the line is reflected over the y-axis
3. If c  0 then the line is translated right by c units
If c  0 then the line is translated left by c units
4. If d  0 then the line is translated up by d units
If d  0 then the line is translated down by d units
1
b
1
b
STATION 2
1. Write the equation of the line that is parallel to y  3x  4 and passes
through the point (5, 8).
2. Write the equation of the line that is perpendicular to y 
3
x  8 and passes
2
through the point (-6, 7).
3. Write the equation of the line that is parallel to 4 x  3 y  9 and has the
same x-intercept as 4 x  6 y  8 .
4. Write the equation of the line that is perpendicular to y  5  2( x  6) and
1
has the same y-intercept as y  4  ( x  12) .
4
STATION 2 ANSWER KEY
1. Parallel lines have the same slope but different y-intercepts. In the equation y = 3x - 4, the slope is 3. So the
equation of our new line will also have a slope of 3. Since the new line has to pass through the point (5, 8), an
equation in point-slope form can be written: y - 8 = 3(x - 5). However, if you want the equation in slope-intercept
form, solve for y: y - 8 = 3(x - 5)
y - 8 = 3x - 15
y = 3x - 7
2. Perpendicular lines have slopes that are opposite reciprocals. In the equation y = (3/2)x -8 the slope is 3/2. The line
that is perpendicular will have a slope of -2/3. Using the given point (-6, 7) an equation in point-slope form can be written:
y - 7 = -2/3(x + 6). Now if you want it in slope-intercept form, solve for y: y - 7 = -2/3(x + 6)
y - 7 = (-2/3)x - 4 y = (-2/3)x + 3
3. To find the slope in a standard form equation: -A/B. In the equation 4x + 3y = -9, A =4 and B = 3, so the slope of this
line would be -4/3. Now find the x-intercept for the equation 4x - 6y = 8 by substituting in y = 0 and solving for x.
4x - 6y = 8
 4x = 8 if y=0
x = 2 Now substitute the new slope and the x-intercept into a point-slope form equation: y - 0 = -4/3(x - 2). This
can be written in slope-intercept form as well by solving for y: y = (-4/3)x + 8/3.
4. In order to have a perpendicular line, the slope must be the opposite reciprocal of the given slope. The given slope
in y-5 = 2(x + 6) is m = 2, so the negative reciprocal is -1/2. Now to find the y-intercept of y + 4 = 1/4(x- 12), solve the
equation for y, this puts the equation in slope-intercept form: y + 4 = 1/4(x - 12)
y + 4 = (1/4)x - 3 y = (1/4)x - 7
Now write the perpendicular line equation using the new slope and y-intercept in point-slope form: y –(-7) = -1/2(x - 0).
Solve for y if you want it in slope-intercept form: y + 7 = -1/2x
y = (-1/2)x - 7
STATION 3
Describe the transformation of the function from the parent linear function
𝑓 (𝑥 ) = 𝑥. Then write the equation for the new function after the
transformation.
1. 𝑔(𝑥 ) = 𝑓 (𝑥 ) + 3
2. ℎ(𝑥 ) = 𝑓(𝑥 − 5)
3. 𝑗(𝑥 ) = −4𝑓(𝑥 + 3)
2
4. 𝑘(𝑥 ) = 𝑓 (− 𝑥) − 5
3
STATION 3 ANSWER KEY
1. The graph of g(x) is translated up 3 units. The new equation is 𝑔(𝑥 ) = 𝑥 + 3.
2. The graph of h(x) is translated right 5 units. The new equation is ℎ(𝑥 ) = 𝑥 − 5.
3. The graph of j(x) is reflected over the x-axis, vertically stretched by a factor of 4,
and translated left 3 units. The new equation is 𝑗(𝑥 ) = −4𝑥 − 12.
3
4. The graph of k(x) is reflected over the y-axis, horizontally stretched by a factor of 2,
2
and translated down 5 units. The new equation is 𝑘(𝑥 ) = − 3 𝑥 − 5.
STATION 4
Describe the transformation of the function when 𝑓 (𝑥 ) = 3𝑥 + 2
1
1. 𝑔(𝑥 ) = − 𝑓 (𝑥 ) − 2
2
2. ℎ(𝑥 ) = 𝑓 (𝑥 + 5) + 4
3. 𝑗(𝑥 ) = 2𝑓 (𝑥 − 3) + 1
4. 𝑘(𝑥 ) = 3𝑓 (−2𝑥 + 4) − 3
STATION 4 ANSWER KEY
1. The graph of g(x) has been reflected over the x-axis, vertically
1
compressed by a factor of , and translated down 2 units. The new
3
2
equation is 𝑔(𝑥 ) = − 𝑥 − 3.
2
2. The graph of h(x) has been translated left by 5 units and translated
up 4 units. The new equation is ℎ(𝑥 ) = 3𝑥 + 21.
3. The graph of j(x) has been vertically stretched by a factor of 2,
translated right by 3 units, and translated up 1 unit. The new
equation is 𝑗(𝑥 ) = 6𝑥 − 13.
4. The graph of k(x) has been vertically stretched by a factor of 3,
1
reflected over the y-axis, horizontally compressed by a factor of ,
2
translated left by 4 units, and translated down by 3 units. The new
equation is 𝑘(𝑥 ) = −18𝑥 + 39.
STATION 5
1. What is the equation of the line that has the same x-intercept as
−3𝑥 + 5𝑦 = 18 and is parallel to the y-axis?
2. What is the equation of the line that has the same y-intercept as
4𝑥 − 5𝑦 = −20 and is parallel to the x-axis?
3. A line has the equation of x = -3
a) Write the equation of the line that is perpendicular to x = -3 and passes
through the point (4, 7).
b) Write the equation of the line that is parallel to x = -3 and passes
through the point (-5, -8).
4. Write the equation of the line that has the same slope as x = 4 and passes
through the same x-intercept as 4𝑥 + 9𝑦 = 15.
STATION 5 ANSWER KEY
1. To find the x-intercept, substitute y = 0 into -3x + 5y =18
-3x = 18
x = -6 (-6, 0)
Any line parallel to the y-axis is a vertical line. So, the equation is x = -6.
2. To find the y-intercept, substitute x = 0 into 4x - 5y = -20
-5y = -20
y = 4 (0, 4)
Any line parallel to the x-axis is a horizontal line. So, the equation is y = 4.
3a. A line that is perpendicular to a vertical line is a horizontal line. The equation of a
horizontal line is y = b. The given point is (4, 7). So, the equation is y = 7.
3b. A line parallel to a vertical line is also vertical. The equation of a vertical line is x = a.
The given point is (-5, -8). So the equation is x = -5.
4. An equation x = 4 has an undefined slope, so a line parallel to this will also have an
undefined slope. The x- intercept is when y = 0. In the equation 4x + 9y = 15, substitute
y = 0 and solve for x.
4x = 15
x = 15/4, so the equation that is parallel to x = 4 is x = 15/4.
STATION 6
ABCD is a square. If the equation of the line going through points C and D is 𝑦 =
3𝑥 − 4, what would be the equation of the line passing through the points B
and C if the coordinates of point B are (3, 5)?
STATION 6 ANSWER KEY
Since ABCD is a square, line segment CD will be perpendicular to line segment
BC. Lines that are perpendicular to each other have slopes that are negative
reciprocals. The slope of line segment CD is 3, therefore the slope for line
1
segment BC will be − .
3
The coordinates of point B are (3, 5), so using the point-slope equation of a line
1
the equation for line segment BC is 𝑦 − 5 = − (𝑥 − 3).
3
To write in slope-intercept form, solve the equation for y.
1
𝑦 − 5 = − (𝑥 − 3)
3
1
𝑦−5=− 𝑥+1
3
1
𝑦 =− 𝑥+6
3
STATION 7
For #1-4, the parent linear function 𝑓 (𝑥 ) = 𝑥 has been transformed. Write the
new function.
1. The graph of g(x) has been horizontally stretched by a factor of 2, translated left by
1
3 units, vertically compressed by a factor of , and translated down 2 units.
4
2. The graph of h(x) has been reflected over the y-axis, horizontally compressed by a
factor of 1/3, vertically stretched by a factor of 4, and translated up 1 unit.
1
3. The graph of j(x) has been horizontally compressed by a factor of 2, translated right
3
1 unit, reflected over the x-axis, vertically compressed by a factor of 5, and
translated down 3 units.
4
4. The graph of k(x) is horizontally stretched by a factor of 3 and is vertically
2
compressed by a factor of .
3
STATION 7 ANSWER KEY
1
1
1 1
1
3
1
5
1. 𝑔(𝑥 ) = 4 ∙ 𝑓 (2 𝑥 + 3) − 2 = 4 (2 𝑥 + 3) − 2 = (8 𝑥 + 4) − 2 = 8 𝑥 − 4
2. ℎ(𝑥 ) = 4 ∙ 𝑓(−3𝑥 ) + 1 = 4(−3𝑥 ) + 1 = −12𝑥 + 1
3
3. 𝑗(𝑥 ) = − 5 ∙ 𝑓(2𝑥 − 1) − 3 =
2
3
2 3
−3
5
6
(2𝑥 − 1) − 3 =
1
4. 𝑘(𝑥 ) = 3 ∙ 𝑓 (4 𝑥) = 3 (4 𝑥) = 12 𝑥 = 2 𝑥
−6
5
3
𝑥+5−3=
−6
5
𝑥−
12
5
STATION 8
Best Buy tracks the number of people who purchase an item at their
website bestbuy.com. The table below shows the number of shoppers per
day during the month of November.
Day
1
3
4
7
8
9
11
15
18
Shoppers 45
92
57
108
65
119 204 234 197
Enter this data into a ‘Lists and Spreadsheets’ page on your calculator. Then
create a scatter plot on a ‘Data and Statistics’ page.
1. Determine the correlation coefficient of the data and interpret the
meaning of this coefficient.
2. Write a regression model for this situation and describe the meaning of the
variables.
STATION 8 ANSWER KEY
1. The correlation coefficient is r = 0.87 which means the data has a strong positive correlation.
More shoppers went online to purchase an item as it became closer to Thanksgiving.
2. The regression model is 𝑦 = 10.9973𝑥 + 31.6873 where x is the number of days into
November and y is the number of shoppers who purchased an item online.