PCM 12 Diagnostic Worksheet.

PCM
12
Diagnostic Worksheet.
1. Draw the next three terms for each pattern shown.
2. Determine whether the two expressions are equivalent.
a. 6n + 8 and 2(3n + 4)
b. (n2 + 4n) – n2 and 4n
c. (5x + 3) + 3x2 – 2 and (5x + 1) + 3x2
d. (7x2 + 1) – (3x – 1)(x + 4) and 2x(x – 3) + 2x2 – 5x + 5
e. (2x + 1)2 – 2x(x – 3) and 6x2 + 6x + 2 – (2x –- 1)2
3. Predict the function family of m(x) and sketch the graph of m(x) using key points on the grid
provided below.
m(x) = f(x) + g(x)
f(x) = 2x; g(x) = –x – 3
4. Draw the function j(x) with outputs such that k(x) = h(x) + j(x) on the grid provided below.
Complete the table of values to verify that h(x) + j(x) = k(x).
5. Algebraically show that h(x) + j(x) is equivalent to k(x).
a. h(x) = –3x + 5;
j(x) = –5x – 7;
b. h(x) = 12x + 9; j(x) = 12x + 6;
k(x) = –8x – 2
k(x) = x + 15
6. Convert the quadratic function in factored form to standard form.
f(x) = (x + 2)(x + 9)
7. Convert the quadratic function in vertex form to standard form.
f(x) = –2(x + 1)2 – 5
8. Write a quadratic function to represent each situation using the given information. Sasha is
training her dog, Bingo, to run across an arched ramp, which is in the shape of a parabola. To
help Bingo get across the ramp, Sasha places a treat on the ground where the arched ramp
begins and one at the top of the ramp. The treat at the top of the ramp is a horizontal distance of
2 feet from the first treat, and Bingo is 6 feet above the ground when he reaches the top of the
ramp. Write a function to represent Bingo’s height above the ground as he walks across the
ramp in terms of his distance from the beginning of the ramp.
9. Circle the function that matches each graph. Explain your reasoning.
10. Circle the function that matches each graph. Explain your reasoning.
11. Given f(x) = x2 , complete the table and graph h(x).
h(x) = (x + 2)2 – 1
12. Given f(x) = x2 , complete the table and graph h(x).
h(x) = x 2 – 9
13. Graph the vertical dilation of f(x) = x2 and tell whether the transformation is a vertical stretch or
a vertical compression and if the graph includes a reflection.
1
g(x) = – x2 – 3
2
14. Use your knowledge of reference points to write an equation for the quadratic function that
satisfies the given information. Use a graph to help solve each problem.
Given: vertex (–4, 2) and y-intercept (0, 10)
16. Write the function that represents the graph.
17. The graph of f(x) is shown. Sketch the graph of the given
transformed function. t(x) = –f(x – 4)
18. Create a quadratic equation and use the set of three given points (–2, –2), (1, –5), (2, –18) that
lie on a parabola. Hint: Use the quadratic equation in STD form: y = ax2 + bx + c
19. Use the vertex form of a quadratic equation to determine whether the zeros of each function are
real or imaginary. Explain how you know.
a)
f(x) = –2(x – 1)2 – 5
b)
f(x) =
3
(x + 4)2 – 6
4
PCM
12
Diagnostic Worksheet.
1. Draw the next three terms for each pattern shown.
2. Determine whether the two expressions are equivalent.
a. 6n + 8 and 2(3n + 4) = 6n + 8
Yes, they are equal.
b. (n2 + 4n) – n2 and 4n
n2 + 4n – n2 = 4n
Yes, they are equal.
c. (5x + 3) + 3x2 – 2 and (5x + 1) + 3x2
Yes, they are equal.
d. (7x2 + 1) – (3x – 1)(x + 4) and 2x(x – 3) + 2x2 – 5x + 5
7x2 + 1 – 3x2 –11x + 4
2x2 – 6x + 2x2 – 5x + 5
4x2 –11x + 5
e. (2x + 1)2 – 2x(x – 3)
4x2 + 4x + 1 – 2x2 + 6
2x2 + 4x + 7
4x2 – 11x + 5
and
Yes, they are equal.
6x2 + 6x + 2 – (2x – 1)2
6x2 + 6x + 2 – 4x2 + 4x – 1
2x2 + 10x + 1
No, they are not equal.
3. Predict the function family of m(x) and sketch the graph of m(x) using key points on the grid
provided below.
m(x) = f(x) + g(x)
f(x) = 2x; g(x) = –x – 3
m(x) = 2x + (–x – 3)
=
x–3
4. Draw the function j(x) with outputs such that k(x) = h(x) + j(x) on the grid provided below.
Complete the table of values to verify that h(x) + j(x) = k(x).
5. Algebraically show that h(x) + j(x) is equivalent to k(x).
a. h(x) = –3x + 5;
j(x) = –5x – 7;
k(x) = –8x – 2
h(x) + j(x) = –3x + 5 + (–5x – 7) = –3x + 5 –5x – 7 = –8x – 2
Yes, they are equal.
b. h(x) = 12x + 9; j(x) = 12x + 6;
k(x) = x + 15
h(x) + j(x) = 12x + 9 + 12x + 6 = 24 x + 15
No, they are not equal.
6. Convert the quadratic function in factored form to standard form.
f(x) = (x + 2)(x + 9)
Foil it.
f(x) = x2 + 2x + 9x + 18 = x2 + 11x + 18
7. Convert the quadratic function in vertex form to standard form.
f(x) = –2(x + 1)2 – 5
Foil and simplify.
f(x) = –2(x2 + 2x + 1) – 5 = –2x2 – 4x – 2 – 5
= –2x2 – 4x – 7
8. Write a quadratic function to represent each situation using the given information. Sasha is
training her dog, Bingo, to run across an arched ramp, which is in the shape of a parabola. To
help Bingo get across the ramp, Sasha places a treat on the ground where the arched ramp
begins and one at the top of the ramp. The treat at the top of the ramp is a horizontal distance of
2 feet from the first treat, and Bingo is 6 feet above the ground when he reaches the top of the
ramp. Write a function to represent Bingo’s height above the ground as he walks across the
ramp in terms of his distance from the beginning of the ramp.
y = a(x – 2)2 + 6, determine “a” by substituting (4, 0).
0 = a(4 – 2)2 + 6 -----> 0 = 4a + 6, a = –
3
2
3
y = – (x – 2)2 + 6
2
9. Circle the function that matches each graph. Explain your reasoning.
a = – 2 because it opens down
and decreases more.
y- intercept is + 7.
10. Circle the function that matches each graph. Explain your reasoning.
11. Given f(x) = x2 , complete the table and graph h(x).
h(x) = (x + 2)2 – 1
12. Given f(x) = x2 , complete the table and graph h(x).
h(x) = x 2 – 9
13. Graph the vertical dilation of f(x) = x2 and tell whether the transformation is a vertical stretch or
a vertical compression and if the graph includes a reflection.
1
g(x) = – x2 – 3
2
It is a vertical compression and the graph
is reflected in x - axis.
14. Use your knowledge of reference points to write an equation for the quadratic function that
satisfies the given information. Use a graph to help solve each problem.
Given: vertex (–4, 2) and y-intercept (0, 10)
y = a(x + 4)2 + 2, determine “a”
by substituting (0, 10).
10 = a(0 + 4)2 + 2 -----> 8 = a 16
1
=a
2
y=
1
(x + 4)2 + 2
2
15. Write the function that represents the graph.
V: ( 5, 1)
concave down,
a = –2
y = –2(x – 5)2 + 1
16. The graph of f(x) is shown. Sketch the graph of the given
transformed function. t(x) = –f(x – 4)
17. Create a quadratic equation and use the set of three given points (–2, –2), (1, –5), (2, –18) that
lie on a parabola. Hint: Use the quadratic equation in STD form: y = ax2 + bx + c
Solve a system of
–2 = a(–2)2 + b(–2) + c --------> –2 = 4a –2b + c ----> (1)
2
–5 = a(1) + b(1) + c --------> –5 = 1a + 1b + c ----> (2) 3 equations with 3 variables.
–18 = a(2)2 + b(2) + c --------> –18 = 4a + 2b + c ----> (3)
(1) – (3) : 16 = –4b , –4 = b.
(1) – 4(2) : 18 = –6b –3c -------->
18 = –6(–4) –3c = 24 –3c ,
–5 = 1a + 1(–4) + 2 --------> –5 = 1a – 4 + 2, –5 = 1a – 2,
Quadratic equation: y = –3x2 – 4x + 2
3c = 24 – 18 =6, c = 2
–3 = 1a
18. Use the vertex form of a quadratic equation to determine whether the zeros of each function are
real or imaginary. Explain how you know.
a)
f(x) = –2(x – 1)2 – 5
V: ( 1, – 5)
This quadratic function has no real roots, so its zeros are imaginary.
The parabola opens down and its vertex is below the x - axis.
b)
f(x) =
3
(x + 4)2 – 6
4
V: ( –4, – 6)
This quadratic function has two real roots, so its zeros are real.
The parabola opens up and its vertex is below the x - axis.