Math  /  Algebra

QuestionFor the quadratic function f(x)=x2+8xf(x)=x^{2}+8 x, answer parts (a) through ( ff ). (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 (4,16)(-4,-16). (Type an ordered pair, using integers or fractions.) What is the equation of the axis of symmetry? The axis of symmetry is x=4x=-4. (Use integers or fractions for any numbers in the equation.) Is the graph concave up or concave down? Concave down Concave up (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 x-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 0,80,-8. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no xx-intercepts. (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 [16,)[-16, \infty). (Type your answer in interval notation.) (e) Determine where the quadratic function is increasing and where it is decreasing.
The function is increasing on the interval \square

Studdy Solution

STEP 1

1. The function given is f(x)=x2+8x f(x) = x^2 + 8x .
2. We need to find various characteristics of this quadratic function, including the vertex, axis of symmetry, concavity, intercepts, graph, domain, range, and intervals of increase and decrease.

STEP 2

1. Determine the vertex and axis of symmetry, and identify concavity.
2. Find the y y -intercept and x x -intercepts.
3. Graph the function using the information from steps 1 and 2.
4. Determine the domain and range of the function.
5. Identify intervals where the function is increasing and decreasing.

STEP 3

To find the vertex of the quadratic function f(x)=x2+8x f(x) = x^2 + 8x , use the formula for the vertex of a parabola x=b2a x = -\frac{b}{2a} . Here, a=1 a = 1 and b=8 b = 8 .
x=82×1=4 x = -\frac{8}{2 \times 1} = -4
Substitute x=4 x = -4 back into the function to find the y y -coordinate of the vertex:
f(4)=(4)2+8(4)=1632=16 f(-4) = (-4)^2 + 8(-4) = 16 - 32 = -16
Thus, the vertex is (4,16)(-4, -16).

STEP 4

The axis of symmetry is the vertical line that passes through the vertex, which is x=4 x = -4 .

STEP 5

Since the coefficient of x2 x^2 is positive (a=1 a = 1 ), the parabola opens upwards, meaning it is concave up.

STEP 6

To find the y y -intercept, set x=0 x = 0 in the function:
f(0)=02+8×0=0 f(0) = 0^2 + 8 \times 0 = 0
So, the y y -intercept is 0 0 .

STEP 7

To find the x x -intercepts, set f(x)=0 f(x) = 0 :
x2+8x=0 x^2 + 8x = 0
Factor the equation:
x(x+8)=0 x(x + 8) = 0
Thus, the x x -intercepts are x=0 x = 0 and x=8 x = -8 .

STEP 8

Using the vertex (4,16)(-4, -16), axis of symmetry x=4 x = -4 , and intercepts (0,0) (0, 0) and (8,0) (-8, 0) , plot these points and sketch the parabola opening upwards.

STEP 9

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

STEP 10

The range of the function is determined by the vertex, since the parabola opens upwards. The minimum value is the y y -coordinate of the vertex, 16-16, so the range is [16,)[-16, \infty).

STEP 11

The function is decreasing on the interval (,4)(-\infty, -4) as it approaches the vertex, and increasing on the interval (4,)(-4, \infty) as it moves away from the vertex.
The function is increasing on the interval (4,)(-4, \infty).

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