QuestionFor the following quadratic function, (a) find the vertex, the axis of symmetry, and the maximum or minimum function value, and (b) graph the function.
Studdy Solution
STEP 1
1. The given function is a quadratic function of the form .
2. The vertex form of a quadratic function is , where is the vertex.
3. The axis of symmetry for a quadratic function in standard form is given by .
4. The function has a minimum value if and a maximum value if .
STEP 2
1. Identify the coefficients , , and .
2. Calculate the vertex of the quadratic function.
3. Determine the axis of symmetry.
4. Find the maximum or minimum value of the function.
5. Graph the function.
STEP 3
Identify the coefficients , , and from the quadratic function :
STEP 4
Calculate the vertex of the quadratic function. The x-coordinate of the vertex is given by the formula:
Substitute back into the function to find the y-coordinate:
Thus, the vertex is .
STEP 5
Determine the axis of symmetry. The axis of symmetry is the vertical line that passes through the vertex:
STEP 6
Find the maximum or minimum value of the function. Since , the parabola opens upwards, and the function has a minimum value at the vertex:
The minimum value is .
STEP 7
Graph the function. To graph :
1. Plot the vertex .
2. Draw the axis of symmetry .
3. Choose additional points on either side of the vertex, such as and , and calculate their corresponding y-values:
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4. Plot these points and .
5. Sketch the parabola through these points, ensuring it is symmetric about the axis of symmetry.
The vertex is , the axis of symmetry is , and the minimum function value is . The graph is a parabola opening upwards with these characteristics.
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