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Math  /  Algebra

QuestionGraph the parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=−2(x+5)2+3f(x)=-2(x+5)^{2}+3

Studdy Solution

STEP 1

1. The function given is a quadratic function in vertex form: f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k .
2. The vertex form of a parabola is f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k , where (h,k)(h, k) is the vertex.
3. The axis of symmetry is a vertical line that passes through the vertex.
4. The domain of a quadratic function is all real numbers.
5. The range depends on whether the parabola opens upwards or downwards.

STEP 2

1. Identify the vertex of the parabola.
2. Determine the axis of symmetry.
3. Determine the domain of the function.
4. Determine the range of the function.
5. Graph the parabola.

STEP 3

Identify the vertex of the parabola from the equation f(x)=−2(x+5)2+3 f(x) = -2(x+5)^2 + 3 .
The vertex form of a parabola is f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k .
From the equation, we can see that h=−5 h = -5 and k=3 k = 3 .
Thus, the vertex is (−5,3)(-5, 3).

STEP 4

Determine the axis of symmetry.
The axis of symmetry for a parabola in vertex form f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k is the vertical line x=h x = h .
Therefore, the axis of symmetry is x=−5 x = -5 .

STEP 5

Determine the domain of the function.
The domain of any quadratic function is all real numbers, so the domain is:
(−∞,∞) (-\infty, \infty)

STEP 6

Determine the range of the function.
Since the coefficient a=−2 a = -2 is negative, the parabola opens downwards.
The maximum value of the function is the y-coordinate of the vertex, which is k=3 k = 3 .
Thus, the range is:
(−∞,3] (-\infty, 3]

STEP 7

Graph the parabola.
1. Plot the vertex (−5,3)(-5, 3).
2. Draw the axis of symmetry x=−5 x = -5 .
3. Since the parabola opens downwards, sketch the parabola opening down from the vertex.
4. Ensure the parabola is symmetric about the axis of symmetry.
The vertex is (−5,3)(-5, 3), the axis of symmetry is x=−5 x = -5 , the domain is (−∞,∞) (-\infty, \infty) , and the range is (−∞,3] (-\infty, 3] .

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