Question

The graph shows a quadratic function from -10 to 10 on the x and y axes. Points on the graph include (-5, 5), (-3, -3), (-2, -4), (-1, -3), and (1, 5).A graph is shown with the x-axis labeled negative 10 to 10 in increments of 1 and the y-axis labeled negative 10 to 10 in increments of 1. A parabola is shown opening upward with vertex left parenthesis negative 2 comma negative 4 right parenthesis and connecting the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 0 right parenthesis.

How many real solutions does the function shown on the graph have?

Answer 1

During a softball game, a batter hits a ball upward from an initial height of 3 feet. The height, in feet, of the softball is given by s (t )= -16 t ^2 + 70 t + 3, where t is time in seconds and t is greater than or equal to 0. Which is closest to the time when the softball will be 50 feet above the ground?

Answer 2

From the graph, we can see that the parabola opens upward and crosses the x-axis at two points (at x = -4 and x = 0). Since the leading coefficient of the quadratic function is positive (since the parabola opens upward), this means that the function has a minimum value and does not touch or cross the x-axis between these two points.

Therefore, the quadratic function has 2 real solutions.

Answer 3

To find the time when the softball will be 50 feet above the ground, we need to set the equation equal to 50 and solve for t:

-16t^2 + 70t + 3 = 50

Rearranging the equation:

-16t^2 + 70t - 47 = 0

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this equation, a = -16, b = 70, and c = -47.

Plugging in these values, we get:

t = (-70 ± √(70^2 - 4(-16)(-47))) / (2(-16))

Simplifying further:

t = (-70 ± √(4900 - 2992)) / (-32)

t = (-70 ± √(1908)) / (-32)

t = (-70 ± 43.7) / (-32)

Now we can calculate the two possible values for t:

t1 = (-70 + 43.7) / (-32) ≈ 1.64
t2 = (-70 - 43.7) / (-32) ≈ 3.481

The closest time when the softball will be 50 feet above the ground is approximately 1.64 seconds.