1.
Functions of the g(x) = a(x – h)^2 can be graphed by applying the appropriate transformations, one at a time, to the graph of
Correct Answer
B. F(x) = x^2
Explanation
The given correct answer, f(x) = x^2, is the original function before any transformations are applied. This can be inferred from the statement "Functions of the g(x) = a(x – h)^2 can be graphed by applying the appropriate transformations, one at a time, to the graph of f(x) = x^2." This implies that f(x) = x^2 is the starting point or base function, and g(x) = a(x – h)^2 is a transformed version of f(x).
2.
List the sequence of steps required to graph the function f(x) = –(x + 4)^2 – 6
Correct Answer
D. Horizontal translation 4 units to the left, reflection in x-axis, vertical translation 6 units down
Explanation
The correct answer is "Horizontal translation 4 units to the left, reflection in x-axis, vertical translation 6 units down". This is because the function f(x) = –(x + 4)^2 – 6 can be transformed by first translating it 4 units to the left, then reflecting it in the x-axis, and finally translating it 6 units down. This sequence of steps will result in the correct graph of the function.
3.
Which function matches the graph?
Correct Answer
B. F(x) = 2(x + 3)^2 – 1
Explanation
The given function f(x) = 2(x + 3)^2 – 1 matches the graph. This can be determined by analyzing the equation. The function has a parabolic shape with a vertex at (-3, -1) and the coefficient 2 in front of the squared term indicates that the parabola opens upwards. The -1 at the end of the equation shifts the graph one unit downwards. Therefore, this equation corresponds to the given graph.
4.
Consider a parabola P that is congruent to y = x^{2}, opens upward, and has vertex (–1, 3). Now find the equation of a new parabola that results if p is reflected in the x-axis and translated 3 units down
Correct Answer
C. Y = – (x + 1)^2
Explanation
The original parabola P is congruent to y = x^2, opens upward, and has vertex (-1, 3). Reflecting the parabola in the x-axis will change the sign of the x-term, resulting in y = -x^2. Translating the parabola 3 units down will change the y-coordinate of the vertex from 3 to 3 - 3 = 0. Therefore, the equation of the new parabola is y = -(x + 1)^2, which matches the given answer.
5.
A stonewashed jean company has determined the cost in dollars (c) per tonne of stones mined is given by c(x) = 0.2(x – 5)^2 + 7, where x is the number of tonnes of stone. How does the vertex of the parabola of the function compare to the vertex of f(x) = x^2
Correct Answer
D. Up 5 units and right 7 units
Explanation
The vertex of the parabola of the function c(x) = 0.2(x – 5)^2 + 7 is obtained by shifting the vertex of the function f(x) = x^2 right by 5 units and up by 7 units. Therefore, the vertex of c(x) is up 5 units and right 7 units compared to the vertex of f(x).
6.
The graphs of y = x^{2} and another parabola are shown below. What is a possible equation for the second parabola?
Correct Answer
A. Y = 2x^2 + 1
Explanation
The graph of y = x^2 is a parabola that opens upwards. The given equation y = 2x^2 + 1 is also a parabola that opens upwards and is shifted vertically upwards by 1 unit compared to y = x^2. Therefore, it is a possible equation for the second parabola.
7.
(4, 16) is a key point on the graph of f(x) = x^{2}. If the parabola is translated horizontally, 3 units left and reflected vertically in the x-axis, what are the coordinates of the image of the key point?
Correct Answer
C. (1, –16)
Explanation
When the parabola is translated horizontally 3 units left, the x-coordinate of the key point (4, 16) will decrease by 3 units to give us (1, 16). Then, when the parabola is reflected vertically in the x-axis, the y-coordinate of the key point will change its sign, resulting in (1, -16). Therefore, the coordinates of the image of the key point are (1, -16).
8.
Sheila threw a ball into the air. The function h(t) = –5(t – 1)^{2} + 5 models the height of the ball h(t), in meters, t seconds after the ball is thrown. Which of the following was not a transformation applied to f(x) = x^{2} to obtain the graph of h(t) = –5(t – 1)^{2} + 5
Correct Answer
D. Vertical compression by a factor of 1/5
Explanation
The graph of h(t) = –5(t – 1)2 + 5 is a vertical compression of the graph of f(x) = x2. This means that the graph of h(t) is narrower and taller than the graph of f(x). The factor of compression is 1/5, which means that the graph of h(t) is compressed vertically by a factor of 1/5 compared to the graph of f(x). Therefore, the correct answer is that a vertical compression by a factor of 1/5 was not applied to f(x) to obtain the graph of h(t).
9.
The graphs of y = x^{2} and another parabola are shown below. What is the first step that should be performed to transform y = x^{2} to the second parabola?
Correct Answer
C. Vertical stretch by a factor of 2
Explanation
To transform the graph of y = x^2 to the second parabola, the first step that should be performed is a vertical stretch by a factor of 2. This means that every y-coordinate of the original parabola will be multiplied by 2, resulting in a steeper and narrower parabola.
10.
What would be the coordinates for C'' if C(-1, -4) was rotated 180° to create C', then C' was reflected over the line y=-x?
Correct Answer
C. (-4, -1)
Explanation
If point C(-1, -4) is rotated 180 degrees around the origin, it will end up at the point C'(1, 4). Then, if C' is reflected over the line y=-x, the x and y coordinates will swap, resulting in the point C''(-4, -1).