Given the following quadratic equation, find;?
a. the vertex
b. the axis of symmetry
c. the intercepts
d. the domain
e. the range
f. the interval where the function is increasing, and
g. the interval where the function is decreasing
h. Graph the function. y= -x^2+2x+8
Which quadratic equation? y=-x^2+2x+8?
y=x^2+2x+8
x = -b/2a gives you the x-value of the vertex. find y by subsituting into the original equation.
axis of symmetry uses the information found above.
Factor your equation and set = to zero to find the intercepts Is there a negative sign in front of the x^2 term. You have it originally, but it is missing in what you typed again.
The domain is all real numbers. This is true any time you have this type of function. We only have to worry about limiting the domain when we have fractions or roots.
You can find your range with the help of the vertex which is a turning point. The y-values will all be greater than or less than the y at the turning point depending on the direction of your graph. I can't help here because I don't know if there is a negative sign in front of the x^2 or not. Remember the range is the y - values.
If you have a u-shaped graph facing up.
It will decrease to the vertex and then increase to the right of the vertex.
If you have a u-shaped graph facing down or upside-down. It will increase to the vertex and then decrease to the right of the vertex.
To graph, use the vertex, the two intercepts. you might try to add another point her two based on that sketch.
To find the answers to the given questions, you can follow these steps:
Step 1: Determine the coefficients of the quadratic equation. In this case, the coefficients are:
a = -1
b = 2
c = 8
a. To find the vertex of the quadratic equation, you can use the formula: x = -b / (2a). Substituting the values, you get:
x = -2 / (2*-1) = -1
To find the y-coordinate of the vertex, substitute the x-value (-1) into the equation:
y = -(-1)^2 + 2(-1) + 8 = -1 - 2 + 8 = 5
Therefore, the vertex is (-1, 5).
b. The axis of symmetry is the vertical line that passes through the vertex. In this case, it is x = -1.
c. To find the intercepts (x and y), set y = 0 and solve for x to find the x-intercepts. To find the y-intercept, set x = 0 and solve for y.
Setting y = 0:
0 = -x^2 + 2x + 8
You can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Factoring is not possible in this case, so let's use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Using the quadratic formula:
x = (-(2) ± √((2)^2 - 4(-1)(8))) / (2(-1))
x = (-2 ± √(4 + 32)) / (-2)
x = (-2 ± √(36)) / (-2)
x = (-2 ± 6) / (-2)
x1 = 4 / (-2) = -2
x2 = -8 / (-2) = 4
Therefore, the x-intercepts are x = -2 and x = 4.
Setting x = 0:
y = -(0)^2 + 2(0) + 8 = 8
Therefore, the y-intercept is y = 8.
d. The domain of the quadratic function is all real numbers because there are no restrictions on the values of x.
e. To find the range, you can analyze the shape of the quadratic function. Since the coefficient of x^2 is negative (-1), the parabola opens downwards. This means the vertex is the maximum point, and the range is (-∞, y-coordinate of vertex] = (-∞, 5].
f. To determine where the function is increasing, you can observe the graph or analyze the quadratic equation. In this case, the coefficient of x^2 is negative, indicating the function is decreasing. Therefore, the function is increasing in the interval (-∞, -1).
g. Similarly, the function is decreasing after the vertex. Therefore, it is decreasing in the interval [-1, +∞).
h. To graph the function, plot the vertex (-1, 5), the x-intercepts (-2, 0) and (4, 0), and the y-intercept (0, 8). Connect these points with a smooth curve. The graph will be a downward-opening parabola.
I hope this explanation helps you understand how to find the answers to the given questions about the quadratic equation y = -x^2 + 2x + 8.