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Math  /  Algebra

QuestionFor the quadratic function f(x)=−x2−2xf(x)=-x^{2}-2 x, answer parts (a) through (f)(f). (a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.
The vertex is (−1,1)(-1,1). (Type an ordered pair, using integers or fractions.) What is the equation of the axis of symmetry? The axis of symmetry is x=−1x=-1. (Use integers or fractions for any numbers in the equation.) Is the graph concave up or concave down? Concave up Concave down (b) Find the yy-intercept and the xx-intercepts, if any.
What is the yy-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The yy-intercept is 0 . (Type an integer or a simplified fraction.) B. There is no yy-intercept.
What is the xx-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The xx-intercept(s) is/are 0,−20,-2. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is/are no xx-intercept(s). (c) Use parts (a) and (b) to graph the function.
Use the graphing tool to graph the function. □\square (d) Find the domain and the range of the quadratic function.
The domain of ff is (−∞,∞)(-\infty, \infty). (Type your answer in interval notation.) The range of ff is (−∞,1](-\infty, 1]. (Type your answer in interval notation.) (e) Determine where the quadratic furfiction is increasing and where it is decreasing.
The function is increasing on the interval □\square . (Type your answer in interval notation.)

Studdy Solution

STEP 1

1. The quadratic function is f(x)=−x2−2x f(x) = -x^2 - 2x .
2. We need to find the vertex, axis of symmetry, concavity, intercepts, graph, domain, range, and intervals of increase/decrease.

STEP 2

1. Determine the vertex and axis of symmetry.
2. Determine the concavity of the graph.
3. Find the y y -intercept and x x -intercepts.
4. Graph the quadratic function.
5. Determine the domain and range.
6. Determine intervals of increase and decrease.

STEP 3

To find the vertex, use the formula for the vertex of a quadratic function f(x)=ax2+bx+c f(x) = ax^2 + bx + c , which is x=−b2a x = -\frac{b}{2a} .
For f(x)=−x2−2x f(x) = -x^2 - 2x , a=−1 a = -1 and b=−2 b = -2 .
Calculate x=−−22(−1)=−1 x = -\frac{-2}{2(-1)} = -1 .
Substitute x=−1 x = -1 into f(x) f(x) to find the y y -coordinate of the vertex: f(−1)=−(−1)2−2(−1)=−1+2=1 f(-1) = -(-1)^2 - 2(-1) = -1 + 2 = 1 .
The vertex is (−1,1)(-1, 1).
The axis of symmetry is x=−1 x = -1 .

STEP 4

The coefficient of x2 x^2 is negative (a=−1 a = -1 ), indicating the graph is concave down.

STEP 5

To find the y y -intercept, set x=0 x = 0 in f(x) f(x) : f(0)=−(0)2−2(0)=0 f(0) = -(0)^2 - 2(0) = 0 . The y y -intercept is 0.
To find the x x -intercepts, solve f(x)=0 f(x) = 0 : −x2−2x=0 -x^2 - 2x = 0 Factor: −x(x+2)=0 -x(x + 2) = 0 So, x=0 x = 0 or x=−2 x = -2 .
The x x -intercepts are 0 0 and −2-2.

STEP 6

Using the vertex (−1,1)(-1, 1), axis of symmetry x=−1x = -1, and intercepts, plot the graph of the quadratic function.

STEP 7

The domain of a quadratic function is always (−∞,∞) (-\infty, \infty) .
The range is determined by the vertex and concavity. Since the graph is concave down and the vertex is (−1,1)(-1, 1), the range is (−∞,1](-\infty, 1].

STEP 8

The function is increasing on the interval to the left of the vertex and decreasing on the interval to the right.
Since the vertex is at x=−1 x = -1 , the function is increasing on (−∞,−1) (-\infty, -1) and decreasing on (−1,∞) (-1, \infty) .
The function is increasing on the interval (−∞,−1) (-\infty, -1) and decreasing on the interval (−1,∞) (-1, \infty) .

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