Sketch the graph of the quadratic function. Indicate the coordinates of the vertex of the graph.
y=−2x2+12x−11
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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
x
y
To sketch the graph of the quadratic function y = -2x^2 + 12x - 11, we can start by finding the coordinates of the vertex.
The vertex of a quadratic function can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic term and linear term, respectively.
In this case, a = -2 and b = 12, so x = -12/(2*(-2)) = -12/(-4) = 3.
To find the corresponding y-coordinate, we substitute this value into the equation:
y = -2(3)^2 + 12(3) - 11
y = -18 + 36 - 11
y = 7.
Therefore, the vertex of the graph is (3, 7).
Using this information, we can sketch the graph.
To sketch the graph of the quadratic function y = -2x^2 + 12x - 11, we can follow these steps:
Step 1: Determine the vertex of the graph.
The vertex of a quadratic function is given by the formula: x = -b/2a.
In this case, a = -2 and b = 12.
So, the x-coordinate of the vertex is: x = (-12) / (2 * (-2)) = (-12) / (-4) = 3.
To find the y-coordinate, substitute this x-value back into the equation:
y = -2(3)^2 + 12(3) - 11
y = -2(9) + 36 - 11
y = -18 + 36 - 11
y = 7.
Therefore, the vertex of the graph is (3, 7).
Step 2: Plot the vertex on the graph.
Locate the point (3, 7) on the coordinate plane.
Step 3: Plot additional points to create the graph.
To plot additional points, you can choose some x-values that are symmetric to the vertex, as well as some other points to estimate the shape of the curve.
For example:
- When x = 2, calculate y:
y = -2(2)^2 + 12(2) - 11
y = -2(4) + 24 - 11
y = -8 + 24 - 11
y = 5
So, the point (2, 5) is plotted.
- When x = 4, calculate y:
y = -2(4)^2 + 12(4) - 11
y = -2(16) + 48 - 11
y = -32 + 48 - 11
y = 5
So, the point (4, 5) is plotted.
You can plot a few more points using the same method, or use the symmetry of the graph to estimate other points.
Step 4: Connect the plotted points to form the graph.
Using a smooth curve, connect the plotted points to form the graph of the quadratic function y = -2x^2 + 12x - 11.
The sketch of the graph with the indicated vertex at (3, 7) is shown above.
To graph the quadratic function y = -2x^2 + 12x - 11, we can follow these steps:
1. Plot a few points on the graph: Since quadratic functions form a U-shaped curve, we can plot a few points to get a sense of the shape. Choose a few values of x and calculate the corresponding y-values using the equation.
Let's choose x = -3, 0, and 3:
For x = -3: y = -2(-3)^2 + 12(-3) - 11 = -2(9) - 36 - 11 = -18 - 36 - 11 = -65
For x = 0: y = -2(0)^2 + 12(0) - 11 = 0 + 0 - 11 = -11
For x = 3: y = -2(3)^2 + 12(3) - 11 = -2(9) + 36 - 11 = -18 + 36 - 11 = 7
So we have three points: (-3, -65), (0, -11), and (3, 7).
2. Plot the points on the graph: On the x-axis, mark the values -3, 0, and 3. On the y-axis, mark the values -65, -11, and 7. Now, plot these points on the graph.
3. Draw the curve: Since it's a quadratic function, we know that it will form a U-shaped curve. Connect the plotted points with a smooth curve.
4. Find the vertex: The vertex of a quadratic function is the highest or lowest point on the graph. We can find the x-coordinate of the vertex using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation.
For our quadratic function y = -2x^2 + 12x - 11:
a = -2
b = 12
Using the formula x = -b/2a:
x = -(12) / (2*(-2)) = -12 / (-4) = 3
So the x-coordinate of the vertex is 3. To find the y-coordinate, substitute this x value into the equation:
y = -2(3)^2 + 12(3) - 11 = -2(9) + 36 - 11 = -18 + 36 - 11 = 7
Therefore, the vertex of the graph is (3, 7).