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
g) the interval where the function is decreasing
h) graph the function
y= -x^2-6x
Y = -x^2 - 6x.
a. h = Xv = -b/2a = 6/-2 = -3
k = -(-3)^2 - 6*-3 = 9
V(h,k)=(-3,9).
b. Axis = h = Xv = -3.
c. Y = -x^2 - 6x = 0.
x(-x-6) = 0
X = 0 = x-Intercept.
-x-6 = 0
X = -6. = X-Intercept.
d. Domain = All real values of X.
e.
f. X = -6 to -3.
g. X = -3 to 0.
h. Use the following points for graphing:
(X,Y)
(-5,5)
(-4,8)
V(-3,9)
(-2,8)
(-1,5).
To find the information for the given quadratic equation y = -x^2 - 6x, let's go step by step:
a) The vertex of a quadratic equation can be found using the formula: x = -b / (2a), where a and b are the coefficients of x^2 and x, respectively. In this case, a = -1 and b = -6. Plugging these values into the formula, we get x = -(-6) / (2*(-1)) = 6 / -2 = -3.
To find the corresponding y-value of the vertex, substitute the x-coordinate (-3) back into the equation: y = -(-3)^2 - 6(-3) = -9 + 18 = 9.
So, the vertex of the quadratic equation is (-3, 9).
b) The axis of symmetry is a vertical line that passes through the vertex. Its equation can be written as x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry is x = -3.
c) The intercepts are the points where the graph of the equation intersects the x-axis (x-intercepts) and the y-axis (y-intercept).
To find the x-intercepts, set y = 0 and solve the equation: -x^2 - 6x = 0. Factoring out an x, we get x(-x - 6) = 0. So, either x = 0 or -x - 6 = 0. Solving for x, we find x = 0 or x = -6. Therefore, the x-intercepts are (0, 0) and (-6, 0).
To find the y-intercept, set x = 0 and solve the equation: y = -(0)^2 - 6(0) = 0. So, the y-intercept is (0, 0).
d) The domain of a quadratic function is all real numbers since quadratic equations are defined for all values of x.
e) To determine the range, we need to find the minimum or maximum value of the quadratic function. In this case, since the coefficient of x^2 is negative, the parabola opens downward, and the vertex represents the maximum point. So, the range is y β€ 9 (all real numbers less than or equal to 9).
f) The interval where the function is increasing is the interval to the left of the vertex. In this case, the function is increasing in the interval (-β, -3].
g) The interval where the function is decreasing is the interval to the right of the vertex. In this case, the function is decreasing in the interval [-3, +β).
h) To graph the function y = -x^2 - 6x, plot the vertex (-3, 9). Then plot the x-intercepts (0, 0) and (-6, 0). Since it is a downward-opening parabola, draw a smooth curve passing through these points. The graph will be a concave downward parabola, with the vertex as the highest point, opening towards the negative y-direction.