Math / Algebra

QuestionUse the vertex and intercepts to sketch the graph of the quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f(x)=x^{2}-2 x-8$

Studdy Solution

STEP 1

1. The function is a quadratic function of the form $f(x) = ax^2 + bx + c$ .
2. The vertex form of a quadratic function is useful for identifying the vertex.
3. The axis of symmetry can be found using the formula $x = -\frac{b}{2a}$ .
4. The domain of a quadratic function is all real numbers.
5. The range of a quadratic function depends on the vertex and the direction of the parabola (upward or downward).

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ .
2. Find the vertex of the parabola.
3. Determine the axis of symmetry.
4. Find the x-intercepts and y-intercept.
5. Sketch the graph using the vertex and intercepts.
6. Determine the domain and range of the function.

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic function $f(x) = x^2 - 2x - 8$ .
Here, $a = 1$ , $b = -2$ , and $c = -8$ .

STEP 4

Find the vertex of the parabola using the vertex formula $x = -\frac{b}{2a}$ .
$x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1$
Substitute $x = 1$ back into the function to find the y-coordinate of the vertex:
$f(1) = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9$
The vertex is $(1, -9)$ .

STEP 5

Determine the axis of symmetry using the x-coordinate of the vertex:
The axis of symmetry is $x = 1$ .

STEP 6

Find the x-intercepts by setting $f(x) = 0$ and solving for $x$ :
$x^2 - 2x - 8 = 0$
Factor the quadratic:
$(x - 4)(x + 2) = 0$
Set each factor to zero:
$x - 4 = 0 \quad \Rightarrow \quad x = 4$ $x + 2 = 0 \quad \Rightarrow \quad x = -2$
The x-intercepts are $x = 4$ and $x = -2$ .
Find the y-intercept by setting $x = 0$ :
$f(0) = (0)^2 - 2(0) - 8 = -8$
The y-intercept is $(0, -8)$ .

STEP 7

Sketch the graph using the vertex $(1, -9)$ , the axis of symmetry $x = 1$ , the x-intercepts $(-2, 0)$ and $(4, 0)$ , and the y-intercept $(0, -8)$ .
The parabola opens upwards because $a = 1 > 0$ .

STEP 8

Determine the domain and range of the function.
The domain of a quadratic function is all real numbers: $(-\infty, \infty)$ .
The range is determined by the vertex and the direction of the parabola. Since the parabola opens upwards and the vertex is at $(1, -9)$ , the range is:
$[-9, \infty)$
The equation of the parabola's axis of symmetry is $x = 1$ . The domain is $(-\infty, \infty)$ and the range is $[-9, \infty)$ .

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QuestionUse the vertex and intercepts to sketch the graph of the quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f(x)=x^{2}-2 x-8$

Studdy Solution

STEP 1

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ .
2. Find the vertex of the parabola.
3. Determine the axis of symmetry.
4. Find the x-intercepts and y-intercept.
5. Sketch the graph using the vertex and intercepts.
6. Determine the domain and range of the function.

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic function $f(x) = x^2 - 2x - 8$ .
Here, $a = 1$ , $b = -2$ , and $c = -8$ .

STEP 4

Find the vertex of the parabola using the vertex formula $x = -\frac{b}{2a}$ .
$x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1$
Substitute $x = 1$ back into the function to find the y-coordinate of the vertex:
$f(1) = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9$
The vertex is $(1, -9)$ .

STEP 5

Determine the axis of symmetry using the x-coordinate of the vertex:
The axis of symmetry is $x = 1$ .

STEP 6

STEP 7

Sketch the graph using the vertex $(1, -9)$ , the axis of symmetry $x = 1$ , the x-intercepts $(-2, 0)$ and $(4, 0)$ , and the y-intercept $(0, -8)$ .
The parabola opens upwards because $a = 1 > 0$ .

STEP 8

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QuestionUse the vertex and intercepts to sketch the graph of the quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=x2−2x−8f(x)=x^{2}-2 x-8f(x)=x2−2x−8

Studdy Solution

STEP 1

STEP 2

1. Identify the coefficients a a a, b b b, and c c c.2. Find the vertex of the parabola.3. Determine the axis of symmetry.4. Find the x-intercepts and y-intercept.5. Sketch the graph using the vertex and intercepts.6. Determine the domain and range of the function.

STEP 3

Identify the coefficients a a a, b b b, and c c c from the quadratic function f(x)=x2−2x−8 f(x) = x^2 - 2x - 8 f(x)=x2−2x−8.Here, a=1 a = 1 a=1, b=−2 b = -2 b=−2, and c=−8 c = -8 c=−8.

STEP 4

STEP 5

Determine the axis of symmetry using the x-coordinate of the vertex:The axis of symmetry is x=1 x = 1 x=1.

STEP 6

STEP 7

Sketch the graph using the vertex (1,−9) (1, -9) (1,−9), the axis of symmetry x=1 x = 1 x=1, the x-intercepts (−2,0) (-2, 0) (−2,0) and (4,0) (4, 0) (4,0), and the y-intercept (0,−8) (0, -8) (0,−8).The parabola opens upwards because a=1>0 a = 1 > 0 a=1>0.

STEP 8

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QuestionUse the vertex and intercepts to sketch the graph of the quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f(x)=x^{2}-2 x-8$

1. Identify the coefficients $a$ , $b$ , and $c$ .
2. Find the vertex of the parabola.
3. Determine the axis of symmetry.
4. Find the x-intercepts and y-intercept.
5. Sketch the graph using the vertex and intercepts.
6. Determine the domain and range of the function.

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic function $f(x) = x^2 - 2x - 8$ .
Here, $a = 1$ , $b = -2$ , and $c = -8$ .

Determine the axis of symmetry using the x-coordinate of the vertex:
The axis of symmetry is $x = 1$ .

Sketch the graph using the vertex $(1, -9)$ , the axis of symmetry $x = 1$ , the x-intercepts $(-2, 0)$ and $(4, 0)$ , and the y-intercept $(0, -8)$ .
The parabola opens upwards because $a = 1 > 0$ .