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Math 30-1, Solved Problems, Homework Help
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Chapter 1, Function Transformation
p 13: q3, q4, q5, q6, q7, q9, q10
p 20: Your Turn
p 31: q16
p 42: q17
Chapter 3, Polynomial Functions
p 134: q8
Chapter 4, Trigonometry
Practice Test:
Chapter 8, Logarithmic Functions
p 425: q20
Solutions for the released 2022 Mathematics 30-1 Practice Diploma Exam:
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For each function, state the values of h and k, the parameters that represent the horizontal and vertical translations applied to y = f (x).
a) y - 5 = f (x)
b) y = f (x) - 4
c) y = f (x + 1)
d) y + 3 = f (x - 7)
e) y = f (x + 2) + 4
Given the graph of y = f (x) and each of the following transformations,
state the coordinates of the image points A', B', C', D' and E', sketch the graph of the transformed function
a) g(x) = f (x) + 3
b) h(x) = f (x - 2)
c) s(x) = f (x + 4)
d) t(x) = f (x) - 2
Describe, using mapping notation, how the graphs of the following functions can be obtained from the graph of y = f (x).
a) y = f (x + 10)
b) y + 6 = f (x)
c) y = f (x - 7) + 4
d) y - 3 = f (x - 1)
Given the graph of y = f (x), sketch the graph of the transformed function. Describe the transformation that can be applied to the graph of f (x) to obtain the graph of the transformed function. Then, write the transformation using mapping notation.
a) r(x) = f (x + 4) - 3
b) s(x) = f (x - 2) - 4
c) t(x) = f (x - 2) + 5
d) v(x) = f (x + 3) + 2
For each transformation, identify the
values of h and k. Then, write the
equation of the transformed function
in the form y - k = f (x - h).
a) f (x) = 1/x , translated 5 units to the left
and 4 units up
b) f (x) = x2, translated 8 units to the right
and 6 units up
c) f (x) = |x|, translated 10 units to the
right and 8 units down
d) y = f (x), translated 7 units to the left
and 12 units down
What vertical translation is applied to y = x^2 if the transformed graph passes through the point (4, 19)?
What horizontal translation is applied to y = x^2 if the translation image graph passes through the point (5, 16)?
The graph of the function y = x^2 is
translated 4 units to the left and 5 units up
to form the transformed function y = g(x).
a) Determine the equation of the function
y = g(x).
b) What are the domain and range of the
image function?
c) How could you use the description of
the translation of the function y = x^2 to
determine the domain and range of the
image function?
The graph of f (x) = |x| is transformed to
the graph of g(x) = f (x - 9) + 5.
a) Determine the equation of the
function g(x).
b) Compare the graph of g(x) to the graph
of the base function f (x).
c) Determine three points on the graph of
f (x). Write the coordinates of the image
points if you perform the horizontal
translation first and then the vertical
translation.
d) Using the same original points from
part c), write the coordinates of the
image points if you perform the vertical
translation first and then the horizontal
translation.
e) What do you notice about the
coordinates of the image points from
parts c) and d)? Is the order of the
translations important?
The graph of the function drawn in red is a translation of the original function drawn in blue. Write the equation of the translated function in the form y - k = f (x - h).
Janine is an avid cyclist. After cycling to a lake and back home, she graphs her distance versus time (graph A).
a) If she left her house at 12 noon, briefly describe a possible scenario for Janine's trip.
b) Describe the differences it would make to Janine's cycling trip if the graph of the function were translated, as shown in graph B.
c) The equation for graph A could be written as y = f(x). Write the function for graph B.
Architects and designers often use translations in their designs. The image shown is from an Italian roadway.
a) Use the coordinate plane overlay with the base semicircle shown to describe the approximate transformations of the semicircles.
b) If the semicircle at the bottom left of the image is defined by the function y = f(x), state the approximate equations of three other semicircles.
Paul is an interior house painter. He determines that the function n = f(A) gives the number of gallons, n, of paint needed to cover an area, A, in square metres. Interpret n = f(A) + 10 and n = f(A + 10) in this context.
The graph of the function y = x^2 is translated to an image parabola with zeros 7 and 1.
The roots of the quadratic equation x2 - x - 12 = 0 are -3 and 4. Determine the roots of the equation (x - 5)2 - (x - 5) - 12 = 0.
a) Given the graph of y = f (x), graph the functions y = -f (x) and y = f (-x). b) Show the mapping of key points on the graph of y = f (x) to image points on the graphs of y = -f (x) and y = f (-x). c) Describe how the graphs of y = -f (x) and y = f (-x) are related to the graph of y = f (x). State any invariant points.
Sketch the graph of f (x) = |x| reflected in each line. a) x = 3 b) y = -2
Two parabolic arches are being built. The first arch can be modelled by the function y = -x^2 + 9, with a range of 0 ≤ y ≤ 9. The second arch must span twice the distance and be translated 6 units to the left and 3 units down. a) Sketch the graph of both arches. b) Determine the equation of the second arch.
A rectangle is inscribed between the x-axis and the parabola y = 9 - x^2 with one side along the x-axis, as shown. a) Write the equation for the area of the rectangle as a function of x. b) Suppose a horizontal stretch by a factor of 4 is applied to the parabola. What is the equation for the area of the transformed rectangle? c) Suppose the point (2, 5) is the vertex of the rectangle on the original parabola. Use this point to verify your equations from parts a) and b).
The graph of the function y = 2x2 + x + 1 is stretched vertically about the x-axis by a factor of 2, stretched horizontally about the y-axis by a factor of _1 3 , and translated 2 units to the right and 4 units down. Write the equation of the transformed function.
Solutions for the released 2022 Mathematics 30-1 Practice Diploma Exam:
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the full set: |
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What is the effect on the graph of the function y = x^2 when the equation is changed to y = (x + 1)^2?
A The graph is stretched vertically.
B The graph is stretched horizontally.
C The graph is the same shape but
translated up.
D The graph is the same shape but
translated to the left.
The graph shows a transformation of the
graph of y = |x|. Which equation models
the graph?
A y + 4 = |x - 6|
B y - 6 = |x - 4|
C y - 4 = |x + 6|
D y + 6 = |x + 4|
If (a, b) is a point on the graph of y = f(x),
which of the following points is on the
graph of y = f(x + 2)?
A (a + 2, b)
B (a - 2, b)
C (a, b + 2)
D (a, b - 2)
The effect on the graph of y = f(x) if it is
transformed to y = 1/4f(3x) is
A a vertical stretch by a factor of 1/4
and a
horizontal stretch by a factor of 3
B a vertical stretch by a factor of 1/4
and a
horizontal stretch by a factor of 1/3
C a vertical stretch by a factor of 4 and a
horizontal stretch by a factor of 3
D a vertical stretch by a factor of 4 and a
horizontal stretch by a factor of 1/3
Which of the following transformations of f(x) produces a graph that has the same y-intercept as f(x)? Assume that (0, 0) is not a point on f(x).
A -9f(x)
B f(x) - 9
C f(-9x)
D f(x - 9)
Solutions for the released 2022 Mathematics 30-1 Practice Diploma Exam:
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a) Use long division to divide
x^2 + 10x - 24 by x - 2. Express the
result in the form
P(x)/(x - a) = Q(x) + R/(x - a).
b) Identify any restrictions on the variable.
c) Write the corresponding statement that
can be used to check the division.
d) Verify your answer.
p 124 q 2, solution
a) Divide the polynomial
3x^4 - 4x^3 - 6x^2 + 17x - 8 by x + 1 using long division. Express the result
in the form P(x)/(x - a) = Q(x) + R/(x - a).
b) Identify any restrictions on the variable.
c) Write the corresponding statement that
can be used to check the division.
d) Verify your answer.
Determine each quotient, Q, using long
division.
a) (x^3 + 3x^2 - 3x - 2) ÷ (x - 1)
b) (x^3 + 2x^2 - 7x - 2) ÷(x - 2)
c) (2w^3 + 3w^2 - 5w + 2) ÷ (w + 3)
d) (9m^3 - 6m^2 + 3m + 2) ÷ (m - 1)
e) (t^4 + 6t^3 - 3t^2 - t + 8) ÷ (t + 1)
f) (2y^4 - 3y^2 + 1) ÷ (y - 3)
Solutions for the released 2022 Mathematics 30-1 Practice Diploma Exam:
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the full set: |
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To save for a new highway tractor, a truck company deposits $11 500 at the end of every 6 months into an account with an annual percentage rate of 5%, compounded semi-annually. Determine the number of deposits needed so that the account has at least $150 000. Use the formula FV = R[(1 + i )^n - 1]/i , where FV is the future value, n is the number of equal periodic payments of R dollars, and i is the interest rate per compounding period expressed as a decimal.
Solutions for the released 2022 Mathematics 30-1 Practice Diploma Exam:
| specific questions | |
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the full set: |
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