Question content area top left
Part 1
Sketch the graph of the quadratic function. Indicate the coordinates of the vertex of the graph.
y=−2x2+4x+3
Question content area bottom left
Part 1
Use the graphing tool to graph the function.
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graph
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Part 1
-15
-12
-9
-6
-3
3
6
9
12
15
-15
-12
-9
-6
-3
3
6
9
12
15
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y
To sketch the graph of the quadratic function y = -2x^2 + 4x + 3, you can use the following steps:
Step 1: Find the x-coordinate of the vertex using the formula x = -b/2a, where a = -2 and b = 4.
x = -4 / (2 * (-2))
x = -4 / (-4)
x = 1
Step 2: Substitute the x-coordinate of the vertex into the equation to find the y-coordinate.
y = -2(1)^2 + 4(1) + 3
y = -2 + 4 + 3
y = 5
So the vertex of the graph is (1, 5).
Step 3: Plot the vertex on the graph.
Step 4: Choose a few x-values on both sides of the vertex and substitute them into the equation to find the corresponding y-values. For example, you can choose x = -2, -1, 0, 2, and 3.
x = -2: y = -2(-2)^2 + 4(-2) + 3 = -8 - 8 + 3 = -13
x = -1: y = -2(-1)^2 + 4(-1) + 3 = -2 - 4 + 3 = -3
x = 0: y = -2(0)^2 + 4(0) + 3 = 0 + 0 + 3 = 3
x = 2: y = -2(2)^2 + 4(2) + 3 = -8 + 8 + 3 = 3
x = 3: y = -2(3)^2 + 4(3) + 3 = -18 + 12 + 3 = -3
Step 5: Plot the points obtained in Step 4 and connect them to form a smooth curve.
The sketch of the graph of the quadratic function y = -2x^2 + 4x + 3 should look like this:
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V(1, 5)
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Note: The vertical axis represents the y-axis and the horizontal axis represents the x-axis. The x-values are marked on the horizontal axis, and the corresponding y-values are marked on the vertical axis.
To sketch the graph of the quadratic function y = -2x^2 + 4x + 3 and find the coordinates of the vertex, we can follow these steps:
Step 1: Identify the coefficients a, b, and c in the quadratic equation y = ax^2 + bx + c. In this case, a = -2, b = 4, and c = 3.
Step 2: Find the x-coordinate of the vertex using the formula x = -b / (2a). Plugging in the values, we get x = -4 / (2 * -2) = 1.
Step 3: Substitute the x-coordinate of the vertex back into the quadratic equation to find the y-coordinate. We substitute x = 1 into y = -2x^2 + 4x + 3 to get y = -2(1)^2 + 4(1) + 3 = -2 + 4 + 3 = 5.
So the coordinates of the vertex are (1, 5).
Now let's use the graphing tool to graph the function:
- Set the x-axis range from -15 to 15.
- Set the y-axis range from -15 to 15.
- Plot the points (-15, y), (-12, y), (-9, y), (-6, y), (-3, y), (3, y), (6, y), (9, y), (12, y), (15, y) to create a smooth curve.
- Indicate the vertex (1, 5) on the graph.
This will give you the graph of the quadratic function y = -2x^2 + 4x + 3.
To sketch the graph of the quadratic function and indicate the coordinates of the vertex, you can follow these steps:
1. Start by plotting the x and y-axis on a graph paper or by using a graphing tool.
2. In the given quadratic function, y = -2x^2 + 4x + 3, we can identify the coefficients for the quadratic term, linear term, and constant term. The coefficient of x^2 is -2, the coefficient of x is 4, and the constant term is 3.
3. To find the coordinates of the vertex, you can use the formula x = -b/2a, where a is the coefficient of x^2 (-2 in this case) and b is the coefficient of x (4 in this case).
Plugging the values into the formula: x = -4 / (2 * -2)
Simplifying: x = -4 / -4
x = 1
4. Now, substitute the value of x (1) into the quadratic function to find the corresponding y-coordinate of the vertex.
Plugging x = 1 into the quadratic function: y = -2(1)^2 + 4(1) + 3
Simplifying: y = -2 + 4 + 3
y = 5
So, the coordinates of the vertex are (1, 5).
5. Plot the vertex on the graph. In this case, the vertex will be a point on the graph with coordinates (1, 5).
6. To sketch the rest of the graph, choose a few x-values, substitute them into the quadratic function to find the corresponding y-values, and plot those points on the graph. You can choose x-values such as -3, -2, 0, 2, and 3.
For example, let's take x = -3:
Plugging x = -3 into the quadratic function: y = -2(-3)^2 + 4(-3) + 3
Simplifying: y = -18 - 12 + 3
y = -27
So, one of the points on the graph is (-3, -27).
7. Repeat step 6 for other x-values and plot the points on the graph.
8. Connect the plotted points smoothly to sketch the graph of the quadratic function.
By using the graphing tool, you can simply input the given quadratic function (y = -2x^2 + 4x + 3) and it will generate the graph for you. You can then identify the vertex on the graph by finding the lowest or highest point on the curve, which represents the vertex.