Question

Find the vertex, axis of symmetry, y intercept, x-intercept(s), whether the parabola opens upwards or downwards, the range (ex: [2, infinity)), the maximum/minimum value, and where the function is increase or decreasing (ex: increasing on: [3, 5], decreasing on [5, 8])


`y=-3/4(x-1/2)^2+9`

Answer 1

Vertex: The vertex of the parabola is at (1/2, 9).

Axis of Symmetry: The axis of symmetry is the vertical line x = 1/2.

Y-Intercept: (0, 143/16).

X-Intercepts: (1/2 + 2√3, 0) and (1/2 - 2√3, 0).

Direction of opening downward.

Range: (-∞, 9], which means the maximum value is 9.

Maximum Value: 9 ( vertex (1/2, 9) )

Increasing/Decreasing Intervals: The function is decreasing on the interval (-∞, 1/2) and increasing on the interval (1/2, ∞).

Explanation:

Vertex: The vertex of the parabola is at (1/2, 9).

Axis of Symmetry: The axis of symmetry is the vertical line x = 1/2.

Y-Intercept: To find the y-intercept, set x to 0 and solve for y:

y = -3/4(0 - 1/2)^2 + 9

y = -3/4(1/4) + 9

y = -3/16 + 9

y = 9 - 3/16

y = 143/16

So, the y-intercept is (0, 143/16).

X-Intercepts: To find the x-intercepts, set y to 0 and solve for x:

0 = -3/4(x-1/2)^2 + 9

-9 = -3/4(x-1/2)^2

Now, divide by -3/4:

-9 / (-3/4) = (x-1/2)^2

12 = (x-1/2)^2

Take the square root of both sides:

√12 = |x-1/2|

x-1/2 = ±√12

x-1/2 = ±2√3

x = 1/2 ± 2√3

So, the x-intercepts are (1/2 + 2√3, 0) and (1/2 - 2√3, 0).

Direction of Opening: The negative coefficient of the squared term (-3/4) means the parabola opens downward.

Range: Since the parabola opens downward, the range is (-∞, 9], which means the maximum value is 9.

Maximum Value: The maximum value of the parabola is 9, which occurs at the vertex (1/2, 9).

Increasing/Decreasing Intervals: The function is decreasing on the interval (-∞, 1/2) and increasing on the interval (1/2, ∞).