QuestionUse the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
What is the vertex?
(Type an ordered pair.)
Studdy Solution
STEP 1
1. The function is a quadratic function.
2. The vertex form of a quadratic function is , where is the vertex.
3. The axis of symmetry for a parabola in standard form is .
STEP 2
1. Find the vertex of the quadratic function.
2. Determine the axis of symmetry.
3. Identify the x-intercepts and y-intercept.
4. Sketch the graph using the vertex and intercepts.
5. Determine the domain and range of the function.
STEP 3
To find the vertex, use the formula for the x-coordinate of the vertex, .
For the quadratic function , we have and .
Calculate the x-coordinate of the vertex:
STEP 4
Substitute back into the function to find the y-coordinate of the vertex:
Thus, the vertex is .
STEP 5
The axis of symmetry is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
STEP 6
To find the y-intercept, set in the function:
So, the y-intercept is .
To find the x-intercepts, solve the equation using the quadratic formula:
The x-intercepts are and .
STEP 7
Sketch the graph using the vertex , the axis of symmetry , the y-intercept , and the x-intercepts and .
STEP 8
The domain of any quadratic function is all real numbers, so the domain is:
The range of the function, since the parabola opens upwards, is all values greater than or equal to the y-coordinate of the vertex:
The vertex of the quadratic function is:
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