How to graph the quadratic function.

Question

In Exercises 5–7, use the following information.

On fourth down, a team is just out of field goal range. The punter is called in to punt. To avoid kicking the ball into the endzone, the punter needs to kick the football high and short. This punt can be modeled by y=􏰁0.088x^2+2.5x+1, where x is the distance (in yards) the foot- ball is kicked and y is the height (in yards) the football is kicked.

5. Graph the quadratic function.

6. Find the greatest height of the football.

7. If the punter punted the ball from the 40-yard line, did the ball reach the endzone?

Answer 1

#5 For the graph you should start by finding the axis of symmetry which would be x = -b/(2a) = -2.5/(2*-0.088) ≈ 14.2. You can then plug in to find the y-coordinate of the vertex, y = -0.088(14.2)²+2.5(14.2)+1 ≈ 18.7. So the vertex of the parabola is (14.2, 18.7) and the parabola opens downward. Another easy point to find would be the y-intercept which is (0, 1) and you can reflect this over the axis of symmetry to get another point (28.4, 1) which should be more than enough for a decent graph.

#6 The greatest height of the football would be the y-value of the vertex, so in this case 18.7 yards.

#7 I assume since the team was "just out of field goal range" that they are on the opposing team's 40 yard line. So this would be 40 yards from the end zone. I'm also going to assume that they mean will the ball reach the endzone on the fly, so we can figure out where the ball will hit the ground and use that to determine the answer. Set the equation equal to zero and solve. You will definitely need to use the quadratic formula.

x = (-2.5±√[(2.5)²-4(-0.088)(1)])/2(-0.088)≈28.8 yards, so from the 40 yard line, the ball would hit the ground at about the 11 yard line and not make it to the endzone (at least without rolling).

Hope this helps!

Answer 2

Hi Tony,

For quadratic equations such as this, there are two main formulas (ahem, "formulae") that you need to know.

A. The first is the quadratic formula: x = (-b +/- √(b² - 4ac)) / 2a

B. The second is the max/min formula: x = -b/2a

We will see how these two formulas come into play in a second. In the meantime, commit these to memory!

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Keeping those two formulas in mind, let's answer the question. All the information you need to use to answer the questions is found in the given equation y = -0.088x² + 2.5x + 1. This is a quadratic equation, y = ax² + bx + c. Quadratic equations form parabolas, which is a fancy word for a symmetrical curve. Many things follow a parabolic path, including football punts.

The "a" term determines whether the parabola opens upwards and has a minimum point ("vertex"), or whether it opens down and has a maximum vertex. If a is positive, it opens up ("U" shape). If a is negative, it opens down ("∩" shape). Even though I don't see a negative sign in the "a" term in your question, the question "greatest" height as well as common knowledge of what punts look like, a must be negative.

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Now that we have context, now let's answer the question. The greatest height (#6) is actually the easiest to answer. The max/min formula indicated earlier shows us the x value for when y is greatest: at point -b/2a. All we have to do is plug in the values: x = - (2.5) / 2(-0.088) = 14.2.

But we want the height, not the horizontal x distance, so we need to plug this newfound x into our quadratic equation: y = max height = -0.088(14.2)² + 2.5(14.2) + 1

max height = 18.74 yards (never forget units!)

Stated simply, the vertex is at (14.2, 18.74).

Now to graph the rest of the parabola. Instead of doing x=1, x=2, and plugging numbers in manually to find points along the parabola, the easiest way is to find the x-intercepts (you can also throw in the y-intercept, where x=0 (y would be 1 here, i.e., the "c" term)).

How do we find x-intercepts? This is where the quadratic formula comes in! x = -b +/- ... when y = 0. The "+/-" term shows us that a parabola cross the x-axis at two points.

Let's just plug these values in:

x = (-(2.5) +/- √(2.5² - 4(-0.088)(1)) / 2(-0.088)

(Quadratic formula calculators can be found online: https://www.calculatorsoup.com/calculators/algebra/quadratic-formula-calculator.php )

Using the + term, x = 28.8

Using the - term, x = -0.39

The quadratic formula thus shows us two new terms: (-0.39, 0) and (28.8, 0).

Plugging these points on a graph, alongside our maximum term that we found earlier, we can sufficiently sketch out the line.

Finally, the quadratic formula also gave us our answer to the final question: if a punter punts from the 40-yard line, will it reach the endzone (0 yard line)? The answer is no! The punt needs to be 40 yards long, but we see that the football hits the ground only 28.8 yards from the place of the punt.

Hope this analysis helps,

JAF