a. The equation represents a parabola. The axis of symmetry is x = 2, the vertex is (2, -4), the y-intercept is (0, 0), the x-intercepts are x = 0 and x = 4, the domain is the set of all real numbers, the range is y ≥ -4, the Increasing Interval is x > 2, the decreasing Interval is x < 2.
b. The equation represents a parabola. The axis of symmetry is x = -3, the vertex is (-3, -14), the y-intercept is (0, -5), the x-intercepts are x = -3 + √14 and x = -3 - √14, the domain is the set of all real numbers, the range is y ≥ -14, the Increasing Interval is x > -3, the decreasing Interval is x < -3.
How to calculate axis of symmetry
a. For the equation y = x² - 4x:
Shape: The equation represents a parabola.
Axis of Symmetry: To find the axis of symmetry, use the formula
x = -b/2a.
In this case, a = 1 and b = -4.
Plug these values into the formula
x = -(-4) / (2 * 1) = 2.
Therefore, the axis of symmetry is x = 2.
To find the vertex, substitute the x-coordinate of the axis of symmetry into the equation to find the corresponding y-coordinate.
Plug x = 2 into the equation
y = (2)² - 4(2) = 4 - 8 = -4.
So the vertex is (2, -4).
To find the y-intercept, substitute x = 0 into the equation.
Plug x = 0,
y = (0)² - 4(0) = 0.
Therefore, the y-intercept is (0, 0).
To find the x-intercepts, set y = 0 and solve for x.
Setting y = 0 in the equation
x² - 4x = 0.
Factoring out x,
x(x - 4) = 0.
So the x-intercepts are x = 0 and x = 4.
Domain: The domain is the set of all real numbers.
Range: The range is y ≥ -4, which means all real numbers greater than or equal to -4.
Increasing Interval: The function is increasing for x > 2.
Decreasing Interval: The function is decreasing for x < 2.
b. For the equation f(x) = x² + 6x - 5:
Shape: The equation represents a parabola.
Axis of Symmetry
use the formula x = -b/2a
In this case, a = 1 and b = 6.
Plug these values into the formula
x = -6 / (2 * 1) = -3.
Therefore, the axis of symmetry is x = -3.
For Vertex:
Substitute x = -3 into the equation
f(-3) = (-3)² + 6(-3) - 5
= 9 - 18 - 5
= -14.
So the vertex is (-3, -14).
For y-intercept:
Substituting x = 0 into the equation
f(0) = (0)² + 6(0) - 5 = -5.
Therefore, the y-intercept is (0, -5).
For x-intercepts:
Setting f(x) = 0, and solve for x.
Solve the quadratic equation
x² + 6x - 5 = 0
You can use factoring, the quadratic formula, or completing the square to find the x-intercepts.
Let's use the quadratic formula:
x = (-6 ± √(6² - 4(1)(-5))) / (2(1)).
x = (-6 ± √(36 + 20)) / 2,
x = (-6 ± √56) / 2.
Therefore, the x-intercepts are x = -3 + √14 and x = -3 - √14.
Domain: The domain is the set of all real numbers.
Range: The range is y ≥ -14, which means all real numbers greater than or equal to -14.
Increasing Interval: The function is increasing for x > -3.
Decreasing Interval: The function is decreasing for x < -3.