i really need help this is very confusing

Question

10.)A woodland jumping mouse hops along a parabolic path given by y = −0.2x^2 + 1.3x,

where x is the mouse’s horizontal distance traveled (in feet) and y is the corresponding height

(in feet). Can the mouse jump over a fence that is 3 feet high? Justify your answer

mathematically!

Answer 1

Hi Liz

y = −0.2x^2 + 1.3x

You can simply calculate the coordinates of the vertex which represents a maximum for the parabola above

a = -0.2

b = 1.3

c = 0

The x coordinate of the vertex

-b/2a =-1.3/2(-0.2) = 1.3/0.4 = 3.25

Plug in this x value to find y

y = -0.2(1.3/0.4)² + 1.3(1.3/0.4)

y = -0.2(1.69/.16) + 1.69/4

y = -1.69/0.8 + 1.69/0.4

y = -1.69 + 3.38/0.8 = 1.69/0.8 = 2.1125

The coordinates of the vertex (3.25, 2.1125) have a y value that is less than 3.

You can graph your parabola at Desmos.com to confirm the coordinates of this maximum for the parabola.

I hope this helps

You can also set your parabola equal to 3 and do the same find the vertex and find the x intercepts using the quadratic formula.

3 = -0.2x² + 1.3x

0 = -0.2x² + 1.3x - 3

Answer 2

This question aims to give you a real world example as to how a parabolic equation could be used to describe motion.

To solve, we need to realize that the 3 foot fence is our y value, so rewording the question, "Is the mouse able to jump a height of 3 or more feet or for what value(s) of x does 3 = - 0.2x² + 1.3x."

A good way to interpret this question is that it is the same as finding the roots of the parabolic path if the path were translated down by 3. The reason this works is because when we find roots, we are finding the possible x values when y=0. So, if instead y=3, we can rewrite the equation to find when a quadratic equals 0 as we are used to. So, rewrite the given equation in the form of 0 = - 0.2x² + 1.3x - 3 and factor.

When we factor this we don't see nice numbers so it's best to use the quadratic formula which is (-b±√b²-4ac)/2a where our quadratic formula is in the standard form ax²+bx+c=0

When you do the quadratic formula there are 3 potential cases that tell you how many roots we have.

If b²-4ac < 0 , then we have no real roots because this gives us imaginary numbers

If b²-4ac = 0, then we have exactly one root

If b²-4ac > 0, then we have two real roots

Once you've determined the case, so long as b²-4ac is not negative, then you have at least one root and therefore the mouse can make the jump. If b²-4ac is negative, then the vertex is lower than y=3 and the mouse cannot make the jump.