Answer:
Part A: the value of h(4) - m(16) is -4
Part B: The y-intercepts are 4 units apart
Part C: m(x) can not exceed h(x) for any value of x
Step-by-step explanation:
Let us use the table to find the function m(x)
There is a constant difference between each two consecutive values of x and also in y, then the table represents a linear function
The form of the linear function is m(x) = a x + b, where
- a is the slope of the function
- b is the y-intercept
The slope = Δm(x)/Δx
∵ At x = 8, m(x) = 2
∵ At x = 10, m(x) = 3
∴ The slope = 
∴ a = 
- Substitute it in the form of the function
∴ m(x) = x + b
- To find b substitute x and m(x) in the function by (8 , 2)
∵ 2 = (8) + b
∴ 2 = 4 + b
- Subtract 4 from both sides
∴ -2 = b
∴ m(x) = x - 2
Now let us answer the questions
Part A:
∵ h(x) = (x - 2)²
∴ h(4) = (4 - 2)²
∴ h(4) = (2)²
∴ h(4) = (4)
∴ h(4) = 2
∵ m(x) = x - 2
∴ m(16) = (16) - 2
∴ m(16) = 8 - 2
∴ m(16) = 6
- Find now h(4) - m(16)
∵ h(4) - m(16) = 2 - 6
∴ h(4) - m(16) = -4
Part B:
The y-intercept is the value of h(x) at x = 0
∵ h(x) = (x - 2)²
∵ x = 0
∴ h(0) = (0 - 2)²
∴ h(0) = (-2)² =
∴ h(0) = 2
∴ The y-intercept of h(x) is 2
∵ m(x) = x - 2
∵ x = 0
∴ m(0) = (0) - 2 = 0 - 2
∴ m(0) = -2
∴ The y-intercept of m(x) is -2
- Find the distance between y = 2 and y = -2
∴ The difference between the y-intercepts of the graphs = 2 - (-2)
∴ The difference between the y-intercepts of the graphs = 4
∴ The y-intercepts are 4 units apart
Part C:
The minimum/maximum point of a quadratic function f(x) = a(x - h) + k is point (h , k)
Compare this form with the form of h(x)
∵ h = 2 and k = 0
∴ The minimum point of the graph of h(x) is (2 , 0)
∵ k is the minimum value of f(x)
∴ 0 is the minimum value of h(x)
∴ The domain of h(x) is all real numbers
∴ The range of h(x) is h(x) ≥ 2
∵ m(8) = 2
∵ m(14) = 5
∵ h(8) = (8 - 2)² = 18
∵ h(14) = (14 - 2)² = 72
∴ h(x) is always > m(x)
∴ m(x) can not exceed h(x) for any value of x
<em>Look to the attached graph for more understand</em>
The blue graph represents h(x)
The green graph represents m(x)
The blue graph is above the green graph for all values of x, then there is no value of x make m(x) exceeds h(x)
, which has derivative 
.
for some value 
within a neighborhood of 
is given by
. Then 
can be estimated to be
![\sqrt[3]{63.97}\approx4-\dfrac{0.03}{48}=3.999375](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B63.97%7D%5Capprox4-%5Cdfrac%7B0.03%7D%7B48%7D%3D3.999375)
for 
, it follows that 