Grade 11 Multiple Transformations To Graph Quiz

1.

Which function matches the graph?

F(x) = –2(x – 3)^2 + 1

F(x) = 2(x + 3)^2 – 1

F(x) = (x + 3)^2 + 2

F(x) = ½ (x – 3)^2 – 1

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.

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.

2. List the sequence of steps required to graph the function f(x) = –(x + 4)^2 – 6

Horizontal translation 4 units to the right, vertical compressions by a factor of 1, vertical translation 6 units down

Horizontal translation 4 units to the right, reflection in the x-axis, vertical translation 6 units down

Horizontal translation 4 units to the left, vertical translation 6 units up, reflection in the x–axis

Horizontal translation 4 units to the left, reflection in x-axis, vertical translation 6 units down

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.

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. 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) = 2x

F(x) = x^2

F(x) = (x – 1)^2

F(x) = x + 2

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).

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).

4. 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) = x² to obtain the graph of h(t) = –5(t – 1)² + 5

Vertical reflection in the x–axis

Vertical translation 5 units up

Horizontal translation 1 unit to the right

Vertical compression by a factor of 1/5

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).

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).

5. Consider a parabola P that is congruent to y = x², 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

Y = –(x + 4)^2 + 3

Y = (x – 1)^2 + 6

Y = – (x + 1)^2

Y = – (x – 2)^2 + 3

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.

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.

6. (4, 16) is a key point on the graph of f(x) = x². 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?

(7, –16)

(–1, 16)

(1, –16)

(–1, –16)

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).

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).

7.

The graphs of y = x² and another parabola are shown below. What is a possible equation for the second parabola?

Y = 2x^2 + 1

Y = ½ x^2 + 1

Y = 2(x + 1)^2

Y = –2x^2 – 1

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.

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.

8. The graphs of y = x² and another parabola are shown below. What is the first step that should be performed to transform y = x² to the second parabola?

Horizontal translation 4 units left

Vertical translation 3 units up

Vertical stretch by a factor of 2

Reflection in x-axis

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.

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.

9. 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

Down 5 units and right 7 units

Up 7 units and left 5 units

Up 7 units and right 5 units

Up 5 units and right 7 units

Explanation

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?

(4, 1)

(1, 4)

(-4, -1)

(-1, -4)

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).

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).