Given the following quadratic equation, find
a. the vertex
b. the axis of symmetry. 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^2-6x
let's complete the square on y = x^2 - 6x
y = x^2 - 6x + 9 - 9
= (x-3)^2 - 9
It is now in the form y = a(x-p)^2 + q
the form you should find in your text or your notes
All your question can now be answered without any further calculations.
To find the answers to the given questions for the quadratic equation y = x^2 - 6x, let's go step by step:
a. Finding the vertex:
The vertex of a quadratic equation can be found using the formula x = -b / (2a). In this equation, a = 1 and b = -6. Plug these values into the formula to find the x-coordinate of the vertex:
x = -(-6) / (2 * 1)
x = 6 / 2
x = 3
To find the y-coordinate of the vertex, substitute the x-coordinate back into the equation:
y = (3)^2 - 6(3)
y = 9 - 18
y = -9
So, the vertex of the quadratic equation is (3, -9).
b. Finding the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is 3, the equation of the axis of symmetry is x = 3.
c. Finding the intercepts:
To find the x-intercepts, set y = 0 and solve the equation:
0 = x^2 - 6x
x(x - 6) = 0
x = 0 or x = 6
So, 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)
y = 0
So, the y-intercept is (0, 0).
d. Finding the domain:
The domain of a quadratic function is all real numbers. So, the domain for this equation is (-∞, ∞).
e. Finding the range:
Since the coefficient of x^2 is positive, the parabola opens upward. This means the lowest point on the parabola is the vertex, which has a y-coordinate of -9. Hence, the range for this equation is [-9, ∞).
f. Finding the interval where the function is increasing:
The function is increasing on the interval (-∞, 3).
g. Finding the interval where the function is decreasing:
The function is decreasing on the interval (3, ∞).
h. Graphing the function:
To graph the equation y = x^2 - 6x, plot the vertex, intercepts, and make a smooth curve that matches the parabolic shape of a quadratic equation. You can use graphing software or a graphing calculator to assist you in this process.
Please note that the graph will be a parabola opening upward with the vertex at (3, -9), the x-intercepts at (0, 0) and (6, 0), and the y-intercept at (0, 0).