Problem 43 - Algebra 2

Question

A model for the height of an arrow shot into the air is \(h(t)=-16 t^{2}+72 t+5\) where \(t\) is time and \(h\) is height. Without graphing, consider the function's graph. a. What can you learn by finding the graph's intercept with the \(h\) -axis? b. What can you learn by finding the graph's intercept(s) with the \(t\) -axis?

Step-by-Step Solution

Answer

The graph's intercept with the h-axis represents the initial height of 5 units. The intercept(s) with the t-axis represent the time(s) when the arrow will hit the ground, which can be found using the quadratic formula.

Step 1: Understanding the h-axis Intercept

The graph's intercept with the h-axis represents the initial height of the arrow when time t is 0. To find this, simply evaluate the function h(t) at t = 0.

Step 2: Calculating the h-axis Intercept

To calculate the h-axis (vertical) intercept, set t = 0 in the function h(t) and solve for h: h(0) = -16(0)^2 + 72(0) + 5.

Step 3: Interpreting the h-axis Intercept Result

Substituting t = 0 in the function yields h(0) = 5. Therefore, the initial height of the arrow is 5 units above the ground level.

Step 4: Understanding the t-axis Intercepts

The graph's intercept(s) with the t-axis represent the time(s) at which the arrow will hit the ground (height will be 0). To find these intercepts, set the function h(t) equal to 0 and solve the resulting quadratic equation for t.

Step 5: Calculating the t-axis Intercepts

Set h(t) equal to 0 and solve for t: 0 = -16t^2 + 72t + 5. This is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula.

Step 6: Applying the Quadratic Formula

Using the quadratic formula, t = [-b ± sqrt(b^2 - 4ac)] / (2a), where a = -16, b = 72, and c = 5, we find the values of t that satisfy the equation.

Step 7: Determining the t-axis Intercepts

Substitute a, b, and c into the quadratic formula to find the t-intercepts: t = [-72 ± sqrt(72^2 - 4*(-16)*5)] / (2*-16).

Step 8: Interpreting the t-axis Intercept Result

Calculate the discriminant (72^2 - 4(-16)5) and evaluate the formula. If the discriminant is positive, there are two real solutions. If it is zero, there is one real solution, and if it is negative, there are no real t-axis intercepts. The solutions give the times when the arrow is at ground level.

Key Concepts

h-axis Intercept

Understanding the h-axis intercept of a quadratic function allows us to find important information about the starting condition of the scenario being modeled. In the context of our arrow model function, given by the equation h(t) = -16t^2 + 72t + 5, the h-axis intercept represents the starting height of the arrow when it is first shot. To calculate this, we evaluate the function at t = 0. Doing so, we find that h(0) = 5, which tells us that when t = 0, the arrow starts at a height of 5 units above the ground level.

It is important to highlight that in a vertical motion problem, such as this, the h-axis intercept can also be interpreted as the initial launch height when no initial velocity is being considered and air resistance is neglected. Hence, interpreting the h-axis intercept provides a foundational piece of information for solving and understanding the motion of objects in a gravitational field.

t-axis Intercept

The t-axis intercepts of a quadratic function indicate the points at which the function's value becomes zero. In practical terms, when analyzing the motion of our arrow given by h(t) = -16t^2 + 72t + 5, identifying the t-axis intercepts allows us to determine the time(s) at which the arrow will reach the ground, i.e., when its height h is zero. We achieve this by setting h(t) to zero and solving for t, resulting in a quadratic equation: 0 = -16t^2 + 72t + 5.

The solutions to this equation are the times at which the arrow intersects with the ground plane. When we solve this using methods like factoring, completing the square, or the quadratic formula, we can find one or two t-axis intercepts. The presence of two intercepts signifies the arrow will reach the ground at two different times, which occurs due to the parabolic nature of its trajectory. The existence of only one t-axis intercept implies the object was released at ground level and returns to it without achieving any height.

Quadratic Formula

The quadratic formula is a cornerstone tool in solving quadratic equations and is written as t = [-b ± sqrt(b^2 - 4ac)] / (2a). It is derived from the process of completing the square and provides a method to determine the roots of any quadratic equation, which can represent t-axis intercepts when pertaining to a function in the form h(t) = at^2 + bt + c.

In the example of our arrow trajectory function h(t) = -16t^2 + 72t + 5, we identify a = -16, b = 72, and c = 5 to apply the quadratic formula. Substituting these values, we calculate the possible times t when the arrow will be at ground level. The term under the square root, known as the discriminant, will determine the nature of the roots; that is, whether we will have real, distinct solutions (the arrow touches ground at different times), one real solution (it grazes the ground), or no real solutions (it never hits the ground in the given context). Applying the quadratic formula is a reliable method to understand and predict the behavior of quadratic functions across various scientific and mathematical disciplines.