Math  /  Algebra

QuestionUse the graph of the quadratic function ff to determine the solution. (a) Solve f(x)>0f(x)>0. (b) Solve f(x)0f(x) \leq 0. (a) The solution to f(x)>0f(x)>0 is \square (Type your answer in interval notation.)

Studdy Solution

STEP 1

1. The quadratic function f(x) f(x) is represented by a parabola opening upwards.
2. The x-intercepts of the parabola are at x=4 x = -4 and x=3 x = 3 .
3. The vertex of the parabola is the lowest point and lies between the x-intercepts.
4. The parabola is symmetric about the vertical line passing through the vertex.

STEP 2

1. Determine the intervals where f(x)>0 f(x) > 0 .
2. Determine the intervals where f(x)0 f(x) \leq 0 .

STEP 3

Identify the intervals where the parabola is above the x-axis. Since the parabola opens upwards, it is above the x-axis outside the x-intercepts. Therefore, f(x)>0 f(x) > 0 for:
x<4andx>3 x < -4 \quad \text{and} \quad x > 3

STEP 4

Express the solution to f(x)>0 f(x) > 0 in interval notation. Since f(x)>0 f(x) > 0 for x<4 x < -4 and x>3 x > 3 , the solution is:
(,4)(3,) (-\infty, -4) \cup (3, \infty)

STEP 5

Identify the intervals where the parabola is on or below the x-axis. Since the parabola opens upwards, it is on or below the x-axis between the x-intercepts, including the intercepts themselves. Therefore, f(x)0 f(x) \leq 0 for:
4x3 -4 \leq x \leq 3

STEP 6

Express the solution to f(x)0 f(x) \leq 0 in interval notation. Since f(x)0 f(x) \leq 0 for 4x3 -4 \leq x \leq 3 , the solution is:
[4,3] [-4, 3]
The solution to f(x)>0 f(x) > 0 is:
(,4)(3,) (-\infty, -4) \cup (3, \infty)
The solution to f(x)0 f(x) \leq 0 is:
[4,3] [-4, 3]

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