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
I will complete the square for you
y = -1(x^2 + 6x+....)
= -(x^2 + 6x + 9 - 9)
= -( (x+3)^2 - 9)
= -(x+3)^2 + 9
I you are studying this topic, then you MUST be able to answer the given questions from this result.
wtf is this?
To find the vertex of the quadratic equation, we can use the formula x = -b/2a. Let's plug in the values from the given equation:
a = -1
b = -6
a) Vertex:
x = -(-6) / (2 * -1) = 6 / -2 = -3
To find the y-coordinate of the vertex, substitute x = -3 into the quadratic equation:
y = -(-3)^2 - 6(-3)
= -9 + 18
= 9
Therefore, the vertex is (-3, 9).
b) Axis of symmetry:
The axis of symmetry is the vertical line that passes through the vertex. Since the vertex has an x-coordinate of -3, we can say that the axis of symmetry is x = -3.
c) Intercepts:
To find the x-intercepts, we set y = 0 and solve for x:
0 = -x^2 - 6x
We can factor out x:
0 = x(-x - 6)
So, the x-intercepts are x = 0 and x = -6.
To find the y-intercept, we set x = 0 and solve for y:
y = -(0^2) - 6(0)
= 0
Therefore, the y-intercept is y = 0.
d) Domain:
The domain of a quadratic equation is the set of all real numbers. So, the domain of this equation is (-∞, +∞).
e) Range:
The given equation is in the form y = -x^2 - 6x. Since the coefficient of x^2 is negative, the parabola opens downwards. Therefore, the range of this equation is (-∞, 9].
f) Interval where the function is increasing:
Since the parabola opens downwards, the function is decreasing on the entire interval.
g) Interval where the function is decreasing:
Since the parabola opens downwards, the function is decreasing on the entire interval.
h) Graphing the function:
To graph the function, plot the vertex (-3, 9) and two other points such as the x-intercepts (0, 0) and (-6, 0). Connect the points with a smooth curve.
The graph of the function y = -x^2 - 6x will be a downward-opening parabola with the vertex at (-3, 9).
(Note: The graph cannot be displayed in a text-based format but can be easily plotted on a graphing calculator or software.)
To find the desired information for the quadratic equation y = -x^2 - 6x, we can follow these steps:
a) The vertex of a quadratic equation in the form y = ax^2 + bx + c can be found using the formula: x = -b / (2a). In this case, a = -1 and b = -6.
Substitute these values into the formula: x = -(-6) / (2(-1)). Simplify: x = 6 / -2 = -3.
The x-coordinate of the vertex is -3. To find the y-coordinate, substitute this value back into the original equation:
y = -(-3)^2 - 6(-3). Simplify: y = -9 + 18 = 9.
Therefore, the vertex is (-3, 9).
b) The axis of symmetry is a vertical line passing through the vertex. In this case, the axis of symmetry is the line x = -3.
c) To find the x-intercepts, set y = 0 in the equation and solve for x:
-x^2 - 6x = 0.
Factor out x: x(-x - 6) = 0.
Set each factor equal to zero and solve for x: x = 0 or -x - 6 = 0.
Solving the second equation: -x = 6, x = -6.
Therefore, the x-intercepts are x = 0 and x = -6.
To find the y-intercept, set x = 0 in the equation: y = -0^2 - 6(0) = 0.
Therefore, the y-intercept is y = 0.
d) The domain of a quadratic function is all real numbers. Therefore, the domain is (-∞, ∞).
e) The range of a quadratic function can be determined by analyzing the vertex.
Since a = -1 is negative, the parabola opens downwards.
Therefore, the maximum value of y occurs at the vertex.
The y-coordinate of the vertex is 9.
Hence, the range is (-∞, 9].
f) To determine the interval where the function is increasing, we need to consider the coefficient of x^2, which is -1.
Since the coefficient is negative, the function is decreasing for all values of x.
Therefore, there is no interval where the function is increasing.
g) Since the coefficient of x^2 is negative, the function is decreasing for all values of x.
Therefore, the interval where the function is decreasing is (-∞, ∞).
h) To graph the function, plot the vertex (-3, 9) on a coordinate plane.
From the vertex, you can also plot the x-intercepts, which are (0, 0) and (-6, 0), and the y-intercept (0, 0).
Draw a smooth curve passing through these points.