find the domain, the range, the axis of symmetry, and the y-intercept, and the x-intercept. (sketch the graph)

Question

Given , find the domain, the range, the axis of symmetry, and the y-intercept, and the x-intercept. (sketch the graph)

Answer 1

The function is f (x ) = 2x - x ^2 + 3.

Let,  f (x ) = y = - x ^2 + 2x + 3.

The function y = - x ^2 + 2x + 3.

Compare the above equation with standard form of parabola i.e, y = ax ^2 + bx + c.

Axis of symmetry x = - b / 2a.

Substitute the values b = 2, and a = - 1.

x = - 2 / 2(- 1) = - 2 / - 2 = 1.

Therefore, axis of symmetry x = 1.

To find the y - intercept, substitute x = 0 in the y = - x ^2 + 2x + 3.

y = - (0)^2 + 2(0) + 3

y = 0 + 0 + 3

y = 3.

To find the x - intercept, substitute y = 0 in the y = - x ^2 + 2x + 3.

0 = - x ^2 + 2x + 3

0 = - x ^2 + 3x - x + 3

0 = - x (x - 3) - 1(x - 3)

0 = (x - 3)(- x - 1)

(x - 3) = 0 and (- x - 1) = 0

x = 3 and x = - 1.

Make the table of values to find ordered pairs that satisfy the equation.

Choose values for x and find the corresponding values for y.

x	y = - x ^2 + 2x + 3	(x, y )
- 2	y = - (- 2)^2 + 2(- 2) + 3 = - 4 - 4 + 3 = - 8 + 3 = - 5	(- 2, - 5)
- 1	y = - (- 1)^2 + 2(- 1) + 3 = - 1 - 2 + 3 = - 3 + 3 = 0	(- 1, 0)
0	y = - (0)^2 + 2(0) + 3 = 0 + 0 + 3 = 3	(0, 3)
1	y = - (1)^2 + 2(1) + 3 = - 1 + 2 + 3 = - 1 + 5 = 4	(1, 4)
2	y = - (2)^2 + 2(2) + 3 = - 4 + 4 + 3 = 3	(2, 3)
3	y = - (3)^2 + 2(3) + 3 = - 9 + 6 + 3 = - 3 + 3 = 0	(3, 0)
4	y = - (4)^2 + 2(4) + 3 = - 16 + 8 + 3 = - 16 + 11 = - 5	(4, - 5)

 

Answer 2

To draw the equation y = - x ^2 + 2x + 3 follow the steps.

1. Draw a coordinate plane.

2. Plot the points on the coordinate plane.

3.Then sketch the graph, connecting the points with a smooth curve.

Since x can be any real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the graph shown.

Notice that every real number is the x - coordinate of some point on the graph, so the domain is all real numbers.

But, only real numbers less than or equal to 4 are y - coordinates of points on the graph. So the range is {y | y ≤ 4}.