3. Use technology to graph the function f(x)=1/2x^2−2x . Then select the true statement below.
A. The vertex and the y-intercept of f(x) are at the same point.
B. The two zeros of f(x) are at the same point
C. The vertex and one of the zeros of f(x) are at the same point.
D. The y-intercept and one of the zeros of f(x) are at the same point.
4. The y-intercept and one of the zeros of f(x) are at the same point. h(t)=−2t^2+6t−4 where h is the height of the dolphin relative to the water level in meters at time t seconds. The graph of h(t) is given below. Then select the true statement.
A. The vertex of h(t) is at (3/2, 1/2) .
B. The two zeros of h(t) are at (−1, 0) and (−2, 0) .
C. The y-intercept of of h(t) is at (−4, 0) .
D. The axis of symmetry of h(t) is t=−3/2
For question 3:
The graph of f(x) = 1/2x^2−2x is a parabola that opens upward. Using technology to graph the function, we can determine the following information:
A. The vertex and the y-intercept of f(x) are at the same point. - Not true. The vertex of the parabola is the lowest point on the graph, while the y-intercept is the point where the graph intersects the y-axis. In this case, the vertex of f(x) is (2, -2) and the y-intercept is (0, 0).
B. The two zeros of f(x) are at the same point - Not true. The zeros of f(x) are the points where the graph intersects the x-axis (y = 0). In this case, the zeros of f(x) are x = 0 and x = 4.
C. The vertex and one of the zeros of f(x) are at the same point. - Not true. As mentioned earlier, the vertex of f(x) is (2, -2) and the zeros of f(x) are x = 0 and x = 4.
D. The y-intercept and one of the zeros of f(x) are at the same point. - Not true. The y-intercept is (0, 0) and the zeros of f(x) are x = 0 and x = 4.
Therefore, none of the statements are true for question 3.
For question 4:
The graph of h(t) = −2t^2+6t−4 is a downward-opening parabola. Using technology to graph the function, we can determine the following information:
A. The vertex of h(t) is at (3/2, 1/2). - Not true. The vertex of the parabola is the highest point on the graph. In this case, the vertex of h(t) is (3/2, 5/2).
B. The two zeros of h(t) are at (−1, 0) and (−2, 0). - Not true. The zeros of h(t) are the points where the graph intersects the x-axis (y = 0). In this case, the zeros of h(t) are t = 1 and t = 2.
C. The y-intercept of h(t) is at (−4, 0). - Not true. The y-intercept is the point where the graph intersects the y-axis. In this case, the y-intercept of h(t) is (0, -4).
D. The axis of symmetry of h(t) is t=−3/2. - Not true. The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry of h(t) is t = 3/2.
Therefore, none of the statements are true for question 4 either.
To graph the function f(x) = 1/2x^2 - 2x, you can use technology such as graphing calculators or graphing software. Here are the steps to do it on a graphing calculator:
1. Turn on your graphing calculator and go to the graphing screen.
2. Enter the function f(x) = 1/2x^2 - 2x. On most calculators, you can use the "x^2" button to enter the squared term.
3. Set up the graphing window by selecting appropriate values for the x-axis and y-axis limits. This will depend on the range you want to see.
4. Choose a suitable scale for the x-axis and y-axis. This will determine how many units represent one increment on the graph.
5. Press the "Graph" button to plot the graph of the function.
6. Once the graph is displayed, you can use the cursor or arrows on the calculator to navigate through the graph and observe different points.
Now let's go through the options for each question:
3. The function f(x) = 1/2x^2 - 2x.
- Option A states that the vertex and the y-intercept of f(x) are at the same point. To check if this is true, you need to find the coordinates of the vertex and the y-intercept.
- The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by (-b/2a, f(-b/2a)). In this case, a = 1/2 and b = -2. Calculate the vertex coordinates.
- The y-intercept is the point where the graph intersects the y-axis. To find it, substitute x = 0 in the function and calculate the value of f(0).
- Compare the coordinates of the vertex and the y-intercept to determine if they are the same point.
- Option B states that the two zeros of f(x) are at the same point. The zeros of a function are the x-values where the function equals zero. To find the zeros, you need to solve the equation f(x) = 0.
- Set up the equation 1/2x^2 - 2x = 0 and solve it for x.
- Check if both solutions are the same point.
- Option C states that the vertex and one of the zeros of f(x) are at the same point. You already determined the coordinates of the vertex and the zeros in the previous steps. Compare them to check if they are the same point.
- Option D states that the y-intercept and one of the zeros of f(x) are at the same point. You already determined the y-intercept and the zeros in the previous steps. Compare them to check if they are the same point.
4. The function h(t) = -2t^2 + 6t - 4.
- Option A states that the vertex of h(t) is at (3/2, 1/2). Similar to the previous question, you can find the coordinates of the vertex using the formula for a quadratic function. In this case, a = -2 and b = 6. Calculate the vertex coordinates and compare them to the given point.
- Option B states that the two zeros of h(t) are at (-1, 0) and (-2, 0). The zeros of h(t) are the values of t where h(t) = 0. Set up the equation -2t^2 + 6t - 4 = 0 and solve for t. Check if both solutions match the given points.
- Option C states that the y-intercept of h(t) is at (-4, 0). The y-intercept is the point where the graph intersects the y-axis. To find it, substitute t = 0 in the function and calculate the value of h(0). Compare the result to the given point.
- Option D states that the axis of symmetry of h(t) is t = -3/2. The axis of symmetry is a vertical line that passes through the vertex of a parabolic function. Check if the equation of the axis of symmetry matches the given value.
By following these steps and calculations, you should be able to determine which statement is true for each question.
3. Use technology to graph the function f(x)=1/2x^2−2x . Then select the true statement below.
A. The vertex and the y-intercept of f(x) are at the same point.
B. The two zeros of f(x) are at the same point
C. The vertex and one of the zeros of f(x) are at the same point.
D. The y-intercept and one of the zeros of f(x) are at the same point.
4. The y-intercept and one of the zeros of f(x) are at the same point. h(t)=−2t^2+6t−4 where h is the height of the dolphin relative to the water level in meters at time t seconds. The graph of h(t) is given below. Then select the true statement.
A. The vertex of h(t) is at (3/2, 1/2) .
B. The two zeros of h(t) are at (−1, 0) and (−2, 0) .
C. The y-intercept of of h(t) is at (−4, 0) .
D. The axis of symmetry of h(t) is t=−3/2
Apologies for the confusion, but I am unable to graph or view images. However, I can still provide some explanations to help you determine the true statements for each question.
For question 3:
To graph the function f(x) = 1/2x^2 - 2x, you can use technology like graphing calculators or graphing software to plot the points and observe the shape of the graph.
A. The statement "The vertex and the y-intercept of f(x) are at the same point" is not true. The vertex of a quadratic function, in this case, is (1, -3), and the y-intercept is (0, 0). They are different points.
B. The statement "The two zeros of f(x) are at the same point" is not true. The zeros, or x-intercepts, of f(x) occur when f(x) = 0. In this case, the zeros are x = 0 and x = 4, which are different points.
C. The statement "The vertex and one of the zeros of f(x) are at the same point" is not true. As mentioned above, the vertex is (1, -3), and the zeros are x = 0 and x = 4, which are different points.
D. The statement "The y-intercept and one of the zeros of f(x) are at the same point" is not true. The y-intercept is (0, 0), and the zeros are x = 0 and x = 4, which are different points.
Therefore, none of the statements are true for question 3.
For question 4:
Since I am unable to view the graph given, I can provide explanations to help you determine the true statement.
A. The statement "The vertex of h(t) is at (3/2, 1/2)" - To find the vertex of h(t) = -2t^2 + 6t - 4, you can determine the x-coordinate by using the formula x = -b/2a, where a = -2 and b = 6. The x-coordinate of the vertex is 1.5 (or 3/2), but we also need to find the y-coordinate. Substituting x = 1.5 into h(t) will give you the y-coordinate. Compare this to the given point to determine if it is true.
B. The statement "The two zeros of h(t) are at (-1, 0) and (-2, 0)" - To find the zeros of h(t), set h(t) = 0 and solve for t. Compare the solutions to the given points to determine if it is true.
C. The statement "The y-intercept of h(t) is at (-4, 0)" - To find the y-intercept of h(t), substitute t = 0 into h(t) and calculate the value. Compare this to the given point to determine if it is true.
D. The statement "The axis of symmetry of h(t) is t = -3/2" - The axis of symmetry of a quadratic function in the form h(t) = at^2 + bt + c is given by t = -b/2a. Compare this to the given equation to determine if it is true.
By analyzing the calculations and comparing the given points, you can determine which statement is true for question 4.
To graph the function f(x) = 1/2x^2 - 2x, you can use a graphing calculator or a graphing software. Here are the steps to graph the function:
1. Open a graphing calculator or graphing software.
2. Enter the function f(x) = 1/2x^2 - 2x into the calculator or software.
3. Set the viewing window to an appropriate range. For example, set the x-axis from -5 to 5 and the y-axis from -10 to 10.
4. Plot the graph of the function by pressing the graph button or equivalent command in the software.
5. The graph should show a parabola opening upward.
Now, let's evaluate the statements for the function f(x) = 1/2x^2 - 2x:
A. The vertex and the y-intercept of f(x) are at the same point: False. The vertex of the parabola is not at the same point as the y-intercept.
B. The two zeros of f(x) are at the same point: False. The zeros of the function (where the graph intersects the x-axis) are not at the same point.
C. The vertex and one of the zeros of f(x) are at the same point: False. The vertex and the zero of the function are at different points.
D. The y-intercept and one of the zeros of f(x) are at the same point: True. The y-intercept of the function is the point where the graph intersects the y-axis, and one of the zeros is where it intersects the x-axis.
For question 4, we will evaluate the statements based on the given graph of the function h(t) = -2t^2 + 6t - 4:
A. The vertex of h(t) is at (3/2, 1/2): False. The vertex of the parabola is not at (3/2, 1/2).
B. The two zeros of h(t) are at (-1, 0) and (-2, 0): False. The zeros of the function (where the graph intersects the x-axis) are not at (-1, 0) and (-2, 0).
C. The y-intercept of h(t) is at (-4, 0): True. The y-intercept is the point where the graph intersects the y-axis, and in this case, it is at (-4, 0).
D. The axis of symmetry of h(t) is t = -3/2: True. The axis of symmetry of a parabola is given by x = -b/2a, where the equation of the parabola is in the form ax^2 + bx + c. In this case, the axis of symmetry is t = -(-6)/(2(-2)) = -3/2.