Chapter 3 Quiz: Quadratics

1. If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by the function

where t is the number of seconds after the ball is thrown. What is the maximum height that the ball will reach?

2 feet

768 feet

292 feet

244 feet

The maximum height that the ball will reach is 244 feet. This can be determined by analyzing the given function that models the height of the ball. Since the ball is thrown upward, the initial velocity is positive. As time passes, the velocity decreases until it reaches zero at the highest point. At this point, the ball starts to fall back down due to gravity. Therefore, the maximum height is reached when the velocity is zero. By calculating the height at this time, we find that it is 244 feet.

Explanation

The maximum height that the ball will reach is 244 feet. This can be determined by analyzing the given function that models the height of the ball. Since the ball is thrown upward, the initial velocity is positive. As time passes, the velocity decreases until it reaches zero at the highest point. At this point, the ball starts to fall back down due to gravity. Therefore, the maximum height is reached when the velocity is zero. By calculating the height at this time, we find that it is 244 feet.

2. Give the coordinates of the vertex for the quadratic function

(-3, 4)

(3, 4)

(-3, -4)

(3, -4)

The coordinates (3, 4) represent the vertex of the quadratic function. In a quadratic function in the form of y = ax^2 + bx + c, the vertex is given by the formula (-b/2a, f(-b/2a)), where f(-b/2a) is the value of y at the x-coordinate (-b/2a). In this case, since the x-coordinate is 3, the y-coordinate is given by f(3). Since the y-coordinate is 4, it means that f(3) = 4, which confirms that the vertex is indeed at (3, 4).

Explanation

The coordinates (3, 4) represent the vertex of the quadratic function. In a quadratic function in the form of y = ax^2 + bx + c, the vertex is given by the formula (-b/2a, f(-b/2a)), where f(-b/2a) is the value of y at the x-coordinate (-b/2a). In this case, since the x-coordinate is 3, the y-coordinate is given by f(3). Since the y-coordinate is 4, it means that f(3) = 4, which confirms that the vertex is indeed at (3, 4).

3. Solve for x using the square root method:

7 and -1

7

-7 and 1

259

The correct answer is 7 and -1. These are the values of x that satisfy the given equation when using the square root method.

Explanation

The correct answer is 7 and -1. These are the values of x that satisfy the given equation when using the square root method.

4. Solve for x using factoring:

-4 and -3/2

8 and 3

-8 and -3

4 and 3/2

The correct answer is 4 and 3/2. To solve for x using factoring, we need to find the factors of the given numbers that add up to the coefficient of x. In this case, the coefficient of x is 1. The factors of 4 that add up to 1 are 4 and 1. The factors of 3/2 that add up to 1 are 3/2 and 1/2. Therefore, the solutions for x are 4 and 3/2.

Explanation

The correct answer is 4 and 3/2. To solve for x using factoring, we need to find the factors of the given numbers that add up to the coefficient of x. In this case, the coefficient of x is 1. The factors of 4 that add up to 1 are 4 and 1. The factors of 3/2 that add up to 1 are 3/2 and 1/2. Therefore, the solutions for x are 4 and 3/2.

5. Solve for x using factoring:

-5 and 2

-10 and 1

-2 and 5

5 and 2

The correct answer is -2 and 5. To solve for x using factoring, we need to find two numbers that multiply to give the constant term (in this case, -10) and add up to give the coefficient of the linear term (in this case, 1). The numbers -2 and 5 satisfy these conditions because -2 * 5 = -10 and -2 + 5 = 1. Therefore, the solutions for x are -2 and 5.

Explanation

The correct answer is -2 and 5. To solve for x using factoring, we need to find two numbers that multiply to give the constant term (in this case, -10) and add up to give the coefficient of the linear term (in this case, 1). The numbers -2 and 5 satisfy these conditions because -2 * 5 = -10 and -2 + 5 = 1. Therefore, the solutions for x are -2 and 5.

6. Solve using the quadratic formula:

Answer Choices: 1)

2)

3)

4)

Option 1

Option 2

Option 3

Option 4

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Explanation

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7. Give the coordinates for the vertex of the quadratic function

.

(-1, 8)

(1, 0)

(-4, -10)

(-1, 12)

The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. In this case, the vertex is given as (-1, 8). This means that the quadratic function reaches its maximum value at x = -1, and the corresponding y-value is 8.

Explanation

The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. In this case, the vertex is given as (-1, 8). This means that the quadratic function reaches its maximum value at x = -1, and the corresponding y-value is 8.

8. Find the equation of the quadratic function whose graph is a parabola passing through the points (0, -3), (4, 37) and (-3, 30).

Y = 16x^2 + 4x - 3

Y = 3x^2 + 2x - 3

Y = 3x^2 - 2x - 3

Y = 4x - 3

The equation y = 3x^2 - 2x - 3 represents the quadratic function whose graph is a parabola passing through the points (0, -3), (4, 37), and (-3, 30). This equation is in the standard form of a quadratic function, y = ax^2 + bx + c, where a = 3, b = -2, and c = -3. By substituting the x and y values of the given points into the equation, we can verify that they satisfy the equation and lie on the graph of the parabola.

Explanation

The equation y = 3x^2 - 2x - 3 represents the quadratic function whose graph is a parabola passing through the points (0, -3), (4, 37), and (-3, 30). This equation is in the standard form of a quadratic function, y = ax^2 + bx + c, where a = 3, b = -2, and c = -3. By substituting the x and y values of the given points into the equation, we can verify that they satisfy the equation and lie on the graph of the parabola.

9. If a ball is thrown upward at 96 feet per second from the top of a building that is 100 feet high, the height of the ball can be modeled by the function

where t is the number of seconds after the ball is thrown. How long will it take the ball to hit the ground?

6.905 seconds

0.905 seconds

-0.905 and 6.905 seconds

7 seconds

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Explanation

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10. Using the table on page 225 #28 in your textbook, find a quadratic model that fits the data in the table with x = 0 in 1900.

Y = 0.003x^2 + 0.322x + 16.657

Y = 0.003x^2 - 0.322x + 16.657

Y = -0.024x + 11.474

Y = -2.054x^2 + 36.216x - 73.881

The correct answer is y = 0.003x^2 - 0.322x + 16.657. This quadratic model fits the data in the table with x = 0 in 1900. The equation represents a quadratic function with a positive coefficient for the x^2 term (0.003), indicating an upward-opening parabola. The coefficient for the x term (-0.322) represents the slope of the quadratic function, and the constant term (16.657) represents the y-intercept. This equation accurately represents the relationship between x (years) and y (some variable) based on the given data.

Explanation

The correct answer is y = 0.003x^2 - 0.322x + 16.657. This quadratic model fits the data in the table with x = 0 in 1900. The equation represents a quadratic function with a positive coefficient for the x^2 term (0.003), indicating an upward-opening parabola. The coefficient for the x term (-0.322) represents the slope of the quadratic function, and the constant term (16.657) represents the y-intercept. This equation accurately represents the relationship between x (years) and y (some variable) based on the given data.