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Math  /  Algebra

Question7) f(x)=−(x+3)2+2f(x)=-(x+3)^{2}+2
Minimum / Maximum Domain: \qquad Range: \qquad x -intercepts: \qquad y-intercept: \qquad Vertex: \qquad

Studdy Solution

STEP 1

1. The function f(x)=−(x+3)2+2 f(x) = -(x+3)^2 + 2 is a quadratic function in vertex form.
2. The vertex form of a quadratic function 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 sign of a a determines whether the parabola opens upwards or downwards.
4. The domain of any quadratic function is all real numbers.
5. The range depends on the direction the parabola opens and the vertex.

STEP 2

1. Determine if the function has a minimum or maximum.
2. Determine the domain of the function.
3. Determine the range of the function.
4. Find the x-intercepts of the function.
5. Find the y-intercept of the function.
6. Identify the vertex of the function.

STEP 3

Determine if the function has a minimum or maximum.
The coefficient of the squared term a=−1 a = -1 is negative, indicating the parabola opens downwards. Therefore, the function has a maximum.

STEP 4

Determine the domain of the function.
The domain of any quadratic function is all real numbers.
Domain: R\mathbb{R}

STEP 5

Determine the range of the function.
Since the parabola opens downwards and has a maximum at the vertex, the range is all values less than or equal to the y-coordinate of the vertex.

STEP 6

Identify the vertex of the function to determine the range.
The function is in vertex form f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k , where the vertex is (h,k)(h, k).
For f(x)=−(x+3)2+2 f(x) = -(x+3)^2 + 2 , the vertex is (−3,2)(-3, 2).
Thus, the range is f(x)≤2 f(x) \leq 2 .
Range: (−∞,2] (-\infty, 2]

STEP 7

Find the x-intercepts of the function.
Set f(x)=0 f(x) = 0 and solve for x x :
0=−(x+3)2+2 0 = -(x+3)^2 + 2 (x+3)2=2 (x+3)^2 = 2 x+3=±2 x+3 = \pm\sqrt{2} x=−3±2 x = -3 \pm \sqrt{2}
The x-intercepts are x=−3+2 x = -3 + \sqrt{2} and x=−3−2 x = -3 - \sqrt{2} .

STEP 8

Find the y-intercept of the function.
Set x=0 x = 0 and solve for f(x) f(x) :
f(0)=−(0+3)2+2 f(0) = -(0+3)^2 + 2 f(0)=−9+2 f(0) = -9 + 2 f(0)=−7 f(0) = -7
The y-intercept is (0,−7) (0, -7) .

STEP 9

Identify the vertex of the function.
As determined earlier, the vertex is (−3,2)(-3, 2).
Summary: - Minimum / Maximum: Maximum - Domain: R\mathbb{R} - Range: (−∞,2] (-\infty, 2] - x-intercepts: x=−3+2,x=−3−2 x = -3 + \sqrt{2}, x = -3 - \sqrt{2} - y-intercept: (0,−7) (0, -7) - Vertex: (−3,2)(-3, 2)

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