x^2-2x-8,=0
graph an appropriate quadratic function
(x-4)(x+2)<=0
x-4<=0 x+2<=0
answer
x<=4 and x,=-2
Yes, that is correct but it says graph ;/.
it looks like you starting statement was
(x-4)(x+2) ≤ 0
there are two critical values on the x-axis. At x=4 and at x=-2
Mark those with a solid dot.
Now you have 3 sections of that graph
1. the part to the left of -2
2. between -2 and 4
3. to the right of 4
So simply test for some value in each of these sections and see if it satisfies the inequation.
If it does, so will every other number in that section.
I found -2 ≤ x ≤ 4
so draw a solid line from -2 to 4 on your graph,
To graph the quadratic function x^2 - 2x - 8 = 0, follow these steps:
1. Start by determining the vertex of the parabola. The formula for the x-coordinate of the vertex is given by -b/2a. In this case, a = 1 and b = -2, so the x-coordinate of the vertex is (-(-2))/(2*1) = 2/2 = 1. To find the y-coordinate, substitute x = 1 back into the equation and solve for y: y = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9. Therefore, the vertex is (1, -9).
2. Next, find the y-intercept by setting x = 0 and solving for y: y = (0)^2 - 2(0) - 8 = 0 - 0 - 8 = -8. So, the y-intercept is (0, -8).
3. To find x-intercepts (if any exist), set y = 0 and solve the equation x^2 - 2x - 8 = 0. You can use factoring, completing the square, or the quadratic formula to solve for x. In this case, the equation can be factored as (x + 2)(x - 4) = 0. Therefore, the x-intercepts are x = -2 and x = 4.
4. Using the vertex, y-intercept, and x-intercepts, you can now sketch the graph of the quadratic function. It will be a downward-facing parabola with the vertex at (1, -9), passing through the points (-2, 0) and (4, 0), and intersecting the y-axis at (0, -8).
Now, let's analyze the inequality (x-4)(x+2) <= 0 step by step:
1. Set each factor to zero and solve individually:
x - 4 = 0 --> x = 4
x + 2 = 0 --> x = -2
2. Create a number line and mark the critical points: -2 and 4.
3. Since the inequality is less than or equal to zero (<= 0), we need to determine where the quadratic is negative or equal to zero.
4. Test points in each of the intervals defined by the critical points:
Choose a point in the interval (-∞, -2): For example, x = -3. Plug it into the inequality: (-3 - 4)(-3 + 2) = (-7)(-1) = 7 > 0. So, this interval is not part of the solution.
Choose a point in the interval (-2, 4): For example, x = 0. Plug it into the inequality: (0 - 4)(0 + 2) = (-4)(2) = -8 < 0. So, this interval is part of the solution.
Choose a point in the interval (4, ∞): For example, x = 5. Plug it into the inequality: (5 - 4)(5 + 2) = (1)(7) = 7 > 0. So, this interval is not part of the solution.
5. The solution to the inequality is the interval (-2, 4] because it includes the points -2 and 4, where the quadratic function is zero or negative.
Therefore, the graph of the appropriate quadratic function and the solution set for the inequality are as follows:
Graph: A downward-facing parabola with the vertex at (1, -9), passing through the points (-2, 0) and (4, 0), and intersecting the y-axis at (0, -8).
Solution set for the inequality: (-2, 4]