Question below: Need explanation. I know how to do the work but I don't really understand it

Question

The equation of the function h is h(x)=1/2(x-2)^2. The table shows some of the values of function m.

———————————————

x ¦ 8 ¦ 10 ¦ 12 ¦ 14 ¦

——————————————

m(x) ¦ 2 ¦ 3 ¦ 4 ¦ 5 ¦

———————————————

Part A: what is the value of h(4)-m(16)?

Part B: If both of these functions are graphed on the same coordinate plane, how far apart are the y-intercepts?

Part C: For what values of x does m(x) exceed h(x)

Answer 1

M is a linear function with slope 1/2

it's intercept is B = y- mx = 2 - 1/2 *8 = 2 - 4 = -2 using (8.2)

so the equation of line M is y = (1/2)x - 2

(a) H(4) - M(16) = 2 - 6 = -4

(b) H(0) = 2 and M(0) = -2, so the difference between the y-intercepts

is 2 - -2 = 4

(c) 1/2(x-2)^2 < (1/2)x - 2

(x-2)^2 < x - 4

x^2 - 4x + 4 < x - 4

x^2 - 5x + 8 < 0

which has no solution. Therefore M(x) < h(x)

Note that h(x) is a parabola with vertex AND minimum at (2,0)

But M(2) = -1, and it never catches up after that.

Answer 2

Part A:

To start off lets take a look at the example values given for us in the table of x vs m(x) values. What the problem does not straight forwardly tell us is that m(x) is linear, however, looking at the growth of x vs m(x) we can see it is directly proportional via a constant in this case 1/2. This is apparent as m(x) grows 1 for every 2 increase in x value.

The equation for a linear function is y = m*x + b, where b is the y intercept, m is the slope which we decided was 1/2, and y in this case is m(x). Plug in any of the values given in the table (ill use x = 8 and y = 2) to get:

2 = 1/2 * 8 + b

b = 2 - 4 = - 2

Thus, m(x) = x/2 - 2

Now that we have both equations we can solve for their difference with the given input values( 4 and 16, respectively)

h(4) - m(16)

[ 1/2 * (4 - 2)² ] - [ 16/2 - 2 ]

[ 1/2 * 2² ] - [ 8 - 2 ]

[ 2 ] - [ 6 ]

-4

Part B:

This part follows quickly after the first as we already know the equation for both functions. The y-intercept is that place that a function intersects the y-axis, in math we also know this line as x=0. With that information, it is clear that if we plug in x=0 to both of our equations, the remaining value will be the place one the y-axis where that function intersects.

h(0) = 1/2 * (0 - 2)²

h(0) = 1/2 * (-2)²

h(0) = 1/2 * 4

h(0) = 2

m(0) = 0/2 - 2

m(0) = 0 - 2

m(0) = -2

Thus, the y-intercepts are 4 units apart.

Part C:

This is probably the trickier of the three problems. Let's think about this graphically. If we want to find all the points where one function is greater than another for the same input, then we first need to find the places where they are the same. Then we can say whether the first function's values are greater or less than the other on either side of those intersecting points.

To do this, we set the equations equal to each other. We can then solve for the x values that make that equation true.

h(x) = m(x)

h(x) - m(x) = 0

[ 1/2 * (x - 2)² ] - [ x / 2 - 2 ] = 0

(x - 2)² - [ x - 4 ] = 0 multiplying both sides by 2

x² - 4x + 4 - x + 4 = 0 multiplying out the square and applying the negative sign to the parentheses

x² - 5x + 8 = 0 simplifying like terms

x = 1/2 (5 ± √-7) using the quadratic we get a solution of non-real numbers

This tells us that there are no points of intersection. So we only need to check one point, which we did in Part B. Thus, we know that h(x) always has greater values than m(x). There are no values of m(x) that exceed h(x).

Answer 3

Hi Rada! I will try to explain the strategy to solve this question. However, I can't tell exactly what concepts are confusing you, so feel free to ask additional questions if I didn't solve your problem.

In order to answer all parts of this question, you first need to figure out what function m is. The only way to do this is to look at the points you are given and make a guess that fits them. Here, I notice that the values for m(x) increase by one each time, so it looks to me like m is a linear function(*). Linear functions have the form ax+b, where a represents the slope and b represents the y-intercept. (Let me know if you would like further explanation on this.)

I notice that x increases by 2 each time m increases by one, so our function m has a slope (change in y / change in x) of 1/2. So, our equation so far is m(x) = (1/2)*x+b.

Now we only need to find b, the y-intercept. The simplest way to do this is to extend the table from the problem down to x=0. Each time x decreases by 2, m(x) will decrease by 1:

_______________

x : 0 : 2 : 4 : 6 : 8 : ...

_______________

m: -2: -1 : 0 : 1 : 2: ...

_______________

So, m's y-intercept is -2.

A more "mathematical" way to do this is by using a point on m to solve our equation, m(x) = (1/2)x+b. Let's try using the point x=8, m(x)=2:

2 = (1/2)*8 + b

2 = 4 + b

-2 = b

Either way, we get that b=-2. So the equation for m is (1/2)*x-2.

With this equation, we can answer part A directly by plugging these values in. For B, you will need to find the

y-intercept of h (which is h(0)). For C, drawing both functions on the same graph (or a graphing calculator like Desmos) should help you find the answer. Look for what part of the graph where m(x) is higher.

I hope this has been helpful! Again, please let me know if there's anything I can further clarify.

Elana

(* If we are being precise, there is more than one function which could fit the points you are given. However, the other functions which fit are not nice to work with, so we will just ignore them.)