Using Technology to Find Rational Factors

8.10.3 Using Technology to Find Rational Factors

8.10.3 • Using Technology to Find Rational Factors

Activity

An engineer is designing a fountain that shoots out drops of water. The nozzle from which the water is launched is 3 meters above the ground. It shoots out a drop of water at a vertical velocity of 9 meters per second.

Function $h$ models the height in meters, $h$ , of a drop of water $t$ seconds after it is shot out from the nozzle. The function is defined by the equation $h (t) = - 5 t^{2} + 9 t + 3$ .

How many seconds until the drop of water hits the ground?

Write an equation that we could solve to answer the question.

Compare your answer:

$- 5 t^{2} + 9 t + 3 = 0$

Try to solve the equation by writing the expression in factored form and using the zero product property. If it cannot be factored, explain how you came to this conclusion.

Compare your answer:

There is no combination of integer factors of the first coefficient, -5, and the third coefficient, 3, that would result in a middle term of 9. Because it is prime, the expression cannot be written in factored form.

Solve the equation by graphing the function.

Use the Desmos graphing tool or technology outside the course.

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Explain how you found the solution.

Compare your answer:

About 2. The graph shows two horizontal intercepts, one with a positive $x$ -coordinate and one negative $x$ -coordinate. The negative one does not apply here because time in seconds cannot be a negative value. The other horizontal intercept is around $(2.087, 0)$ . This means it takes about 2 seconds for the drop to hit the ground.

A graph of a parabola opening downwards, crossing the x-axis at approximately x equals negative 0.29 and x equals 2.09, marked by blue and green points labeled (negative 0.287, 0) and (2.087, 0), respectively.

Using Technology to Find the Rational Factors

Watch the following video to learn more about using technology to find the rational factors of a quadratic equation.

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Self Check

A tennis ball is hit by a racket.

The function $h(t) = -3t^2+ 7t + 3$ models the height in meters, $h$ , of the tennis ball $t$ seconds after the impact with the racket.

Use graphing technology to determine the amount of time that has passed from the tennis ball being hit to when it reaches the ground.

3.124 seconds
0.37 seconds
2.703 seconds
1.167 seconds

Additional Resources

Using Technology to Find Rational Factors

Let’s use graphing technology to solve a situation involving a quadratic function that cannot be solved using factoring.

A thick steel support cable at a construction site forms an arch-like shape. It starts attached to the top of a column, goes up through a loop at a high point, and then back down to the ground.

A black cable hangs in a curve between a pole on the left and a higher point above, forming a catenary shape above a brown ground. The pole resembles a short column or post.

The height of the cable in meters is represented by the function $h (x) = - 4 x^{2} + 16 x + 5$ where $x$ is the horizontal distance from the column, in meters.

How far away from the column does the cable attach to the ground?

The first step in solving would be to set the function equal to 0 and try to factor. Since this equation cannot be factored, we will use graphing technology.

Let’s graph the function using the equation $y = - 4 x^{2} + 16 x + 5$ .

Graph of a downward parabola with a positive x-intercepts at 4.291.

We can see the function has a zero at $x = 4.291$ . The $x$ -intercept is $(4.291, 0)$ .

This means the cable attaches to the ground 4.291 meters away from the column.

What else about the steel cable system do we know from this graph?

We can also tell that the cable attaches to the column 5 meters off the ground at the $y$ -intercept and the loop is approximately 21 meters high at the vertex of the parabola.

Try it

Using Technology to Find Rational Factors

The height, in meters, of a model rocket being launched is modeled by the function $h (t) = - 2 t^{2} + 14 t + 2$ where $t$ is the number of seconds after launch.

How many seconds pass before the rocket reaches the ground?

Here is how to solve this problem using graphing technology:

Since the expression cannot be factored, it should be graphed.

Graph of a downward parabola with a positive x-intercepts at 7.14.

The function has a zero at $x = 7.14$ . The $x$ -intercept is $(7.14, 0)$ .

This means the rocket travels for 7.14 seconds before reaching the ground.