Grade 11 Graphing Quadratic Functions Quiz

1) Kevin threw a ball straight up with an initial speed of 20 metres of second. The function y = –5(x – 2)^2 + 20 describes the ball’s height, in metres, t seconds after Kevin threw it. What are the coordinates of the vertex?

(–5,2)

(2,20)

(20,2)

(–5,20)

The vertex of a quadratic function in the form y = a(x-h)^2 + k is (h,k). In this case, the function is y = –5(x – 2)^2 + 20, so the vertex is (2, 20).

Explanation

The vertex of a quadratic function in the form y = a(x-h)^2 + k is (h,k). In this case, the function is y = –5(x – 2)^2 + 20, so the vertex is (2, 20).

2) Which correctly identifies the values of the parameters a, h, and k for the function f(x) = –2(x + 3)^2 + 1

A = –2, h = 3, k = 1

A = 2, h = –3, k =–1

A = –2, h = –3, k = 1

A = –2, h = –3, k = –1

The given function is in the form f(x) = a(x - h)^2 + k, where a represents the vertical stretch/compression, h represents the horizontal shift, and k represents the vertical shift. In the given function f(x) = -2(x + 3)^2 + 1, a = -2 represents a vertical compression by a factor of 2, h = -3 represents a horizontal shift to the left by 3 units, and k = 1 represents a vertical shift upward by 1 unit.

Explanation

The given function is in the form f(x) = a(x - h)^2 + k, where a represents the vertical stretch/compression, h represents the horizontal shift, and k represents the vertical shift. In the given function f(x) = -2(x + 3)^2 + 1, a = -2 represents a vertical compression by a factor of 2, h = -3 represents a horizontal shift to the left by 3 units, and k = 1 represents a vertical shift upward by 1 unit.

3) Which function includes a translation of 3 units left?

F(x) = (x + 3)^2 + 1

F(x) = 3x^2 + 1

F(x) = (x – 3)^2 + 1

F(x) = (x + 1)^2 – 3

The function f(x) = (x + 3)^2 + 1 includes a translation of 3 units to the left. This can be seen by the term (x + 3) in the function, which shifts the graph of the function 3 units to the left. The square term ensures that the shift is applied to the entire function. The +1 term does not affect the translation, but rather shifts the graph 1 unit up. Therefore, the correct answer is f(x) = (x + 3)^2 + 1.

Explanation

The function f(x) = (x + 3)^2 + 1 includes a translation of 3 units to the left. This can be seen by the term (x + 3) in the function, which shifts the graph of the function 3 units to the left. The square term ensures that the shift is applied to the entire function. The +1 term does not affect the translation, but rather shifts the graph 1 unit up. Therefore, the correct answer is f(x) = (x + 3)^2 + 1.

4) The formula for compound interest on $1000at a given interest rate, r, is shown by the function f(r) = 1000(r + 1)^2. What are the values of the parameters a, h, and k for f(r)

A = 1000, h = 1, k = 0

A = 1000, h = 0, k = 1

A = 0, h = 1000, k = 1

A = 1000, h = –1, k = 0

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Explanation

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5)

What is the equation of this graph

Y = –x^2 + 3

Y = –3x^2

Y = –(x + 3)^2

Y = –(x – 3)^2

The graph is a downward-facing parabola with the vertex at (3,0). The equation y = -(x - 3)^2 represents a parabola that is reflected vertically and shifted 3 units to the right. This matches the graph shown.

Explanation

The graph is a downward-facing parabola with the vertex at (3,0). The equation y = -(x - 3)^2 represents a parabola that is reflected vertically and shifted 3 units to the right. This matches the graph shown.

6) Which equation describes a parabola that opens downward, is congruent to y = x^2, and has its vertex at (0, 3)

Y = (x + 3)^2 – 1

Y = –x^2 + 3

Y = –(x – 3)^2

Y = x^2 + 3

The equation y = -x^2 + 3 describes a parabola that opens downward because the coefficient of x^2 is negative. The vertex of the parabola is at (0, 3) because the constant term in the equation represents the y-coordinate of the vertex. Additionally, the equation is congruent to y = x^2 because it has the same general form, but with a negative coefficient for x^2.

Explanation

The equation y = -x^2 + 3 describes a parabola that opens downward because the coefficient of x^2 is negative. The vertex of the parabola is at (0, 3) because the constant term in the equation represents the y-coordinate of the vertex. Additionally, the equation is congruent to y = x^2 because it has the same general form, but with a negative coefficient for x^2.

7) Leila dropped a pebble from a bridge that is 100 meters above the river. The function h(t) = –5t^2 + 100 describes the height of the pebble, h(t), in meters, t seconds after the pebble is dropped. Which is the axis of symmetry for the graph of the function?

X = –5

X = 0

X = 5

X = 100

The axis of symmetry for a quadratic function in the form of h(t) = -5t^2 + 100 is given by the equation x = -b/2a. In this case, a = -5 and b = 0, so the equation becomes x = -0/2(-5), which simplifies to x = 0. Therefore, the axis of symmetry for the graph of the function is x = 0.

Explanation

The axis of symmetry for a quadratic function in the form of h(t) = -5t^2 + 100 is given by the equation x = -b/2a. In this case, a = -5 and b = 0, so the equation becomes x = -0/2(-5), which simplifies to x = 0. Therefore, the axis of symmetry for the graph of the function is x = 0.

8) Joanne hit a ball straight up into the air. The height of the ball in meters, is given by the function h(t) = –5(t – 3)^2 + 45 t seconds after the ball is hit. In how many seconds will the ball hit the ground?

3

6

9

45

The ball will hit the ground when the height is 0. To find this, we set the function h(t) equal to 0 and solve for t. -5(t - 3)^2 + 45t = 0.

Now, isolate the variable t:

5(t - 3)^2 = 45

(t - 3)^2 = 9

Take the square root of both sides:

t - 3 = ±√9

t - 3 = ±3

Now, solve for t:

t - 3 = 3 t = 3 + 3 t = 6 seconds

t - 3 = -3 t = 3 - 3 t = 0 seconds

Since the ball is hit at t = 0, we only consider positive values of t. Thus, the ball will hit the ground in 6 seconds.

Explanation

The ball will hit the ground when the height is 0. To find this, we set the function h(t) equal to 0 and solve for t. -5(t - 3)^2 + 45t = 0.

Now, isolate the variable t:

5(t - 3)^2 = 45

(t - 3)^2 = 9

Take the square root of both sides:

t - 3 = ±√9

t - 3 = ±3

Now, solve for t:

t - 3 = 3 t = 3 + 3 t = 6 seconds

t - 3 = -3 t = 3 - 3 t = 0 seconds

Since the ball is hit at t = 0, we only consider positive values of t. Thus, the ball will hit the ground in 6 seconds.

9) Consider a parabola P that is congruent to y = x^2 and with vertex at (0, 0). Find the equation of a new parabola that results if P is translated 3 units

Y = (x + 3)^2 + 4

Y = (x – 3)^2 + 4

Y = (x – 4)^2 + 3

Y = (x – 3)^2 – 4

The given equation y = (x - 3)^2 + 4 represents a parabola that has been translated 3 units to the right from the original parabola y = x^2. The term (x - 3)^2 indicates that the vertex of the new parabola is at (3, 0), which is 3 units to the right of the original vertex at (0, 0). The constant term +4 indicates that the new parabola has been shifted upward by 4 units compared to the original parabola. Therefore, the equation y = (x - 3)^2 + 4 represents the equation of the new parabola resulting from the translation.

Explanation

The given equation y = (x - 3)^2 + 4 represents a parabola that has been translated 3 units to the right from the original parabola y = x^2. The term (x - 3)^2 indicates that the vertex of the new parabola is at (3, 0), which is 3 units to the right of the original vertex at (0, 0). The constant term +4 indicates that the new parabola has been shifted upward by 4 units compared to the original parabola. Therefore, the equation y = (x - 3)^2 + 4 represents the equation of the new parabola resulting from the translation.

10) Which equation shows a translation of 3 left and vertical compression by a factor of 2 to the graph y = x^2

Y = 2(x – 3)^2

Y = 2(x + 3)^2

Y = ½(x – 3)^2

Y = ½(x + 3)^2

The equation y = ½(x + 3)^2 shows a translation of 3 units to the left and a vertical compression by a factor of 2 to the graph y = x^2. The translation of 3 units to the left is represented by the term (x + 3), which shifts the graph horizontally. The vertical compression by a factor of 2 is represented by the coefficient ½, which compresses the graph vertically. Therefore, the given equation satisfies both the translation and compression requirements.

Explanation

The equation y = ½(x + 3)^2 shows a translation of 3 units to the left and a vertical compression by a factor of 2 to the graph y = x^2. The translation of 3 units to the left is represented by the term (x + 3), which shifts the graph horizontally. The vertical compression by a factor of 2 is represented by the coefficient ½, which compresses the graph vertically. Therefore, the given equation satisfies both the translation and compression requirements.