R(x)=-x²+5x+6

Question

1.)domain in interval notation? 2.)range of the function in interval notation? 3.)x intercepts, y intercepts, and vertex?

Answer 1

1) The function R(x) = - x² + 5x + 6

Change R(x) to y.

The quadratic function  y = - x² + 5x  + 6 is represents down ward parabola in the graph.

We know that domain of the function is all possible x values.

Parabola domain x = all real numbers.

Domain in interval notation (-∞, ∞).

Answer 2

2) The function R(x) = - x² + 5x + 6

Change R(x) to y.

The quadratic function  y = - x² + 5x  + 6

Compare it to y = ax²  + bx  + c

a = - 1, b = 5, c = 6

In this case a is negative means the parabola opens down.

When the parabola opens down and it has a maximum point which is the vertex of parabola.

To find x coordinate of vertex x = -b/2a

x = - 5/2(- 1)

x = 5/2

To find y coordinate of vertex substitute x = 5/2 in y = - x² + 5x  + 6.

y =  - (5/2)² + 5(5/2)  + 6

y = - (25/4) + 25/2 + 6

y = 49/4

When the parabola opens down it has a maximum point which is the vertex of parabola [(5/2), (49/4)]

We know that range is all possible y values.

In the maximum point y = 49/4  so the graph of parabola cannot be upper than 49/4.

Thus the range of function y ≤ 49/4.

Range of the function is  {y |y ≤ 49/4}

Range in interval notation (-∞, 49/4].

 

Answer 3

3) The function R(x) = - x² + 5x + 6

Change R(x) to y.

The quadratic function  y = - x² + 5x  + 6

To find x intercepts substitute y = 0 in y = - x² + 5x  + 6.

- x² + 5x + 6 = 0

x² - 5x - 6 = 0

x² - 6x + x - 6 = 0

x(x - 6) + 1(x - 6) = 0

(x - 6)(x + 1) = 0

x - 6 = 0 and x + 1 = 0

x = 6 and x = - 1

x intercepts are -1 and 6.

 

To find y intercepts substitute x = 0 in y = - x² + 5x  + 6.

y = - (0)² + 5(0) + 6

y = 6

y intercepts are 6.

 

 y = - x² + 5x  + 6

Compare it to y = ax²  + bx  + c

a = - 1, b = 5, c = 6

To find x coordinate of vertex x = -b/2a

x = - 5/2(- 1)

x = 5/2

To find y coordinate of vertex substitute x = 5/2 in y = - x² + 5x  + 6.

y =  - (5/2)² + 5(5/2)  + 6

y = - (25/4) + 25/2 + 6

y = 49/4

Vertex (x, y) = (5/2, 49/4).