Answer:
Explanation:
Using the absolute value function, linear functions and translations, we have that:
9. Vertex (2, 0), Range: y
10. The equation is f(x) = -5x + 100.
11. The correct option is: The slope of f(x) is greater than the slope of g(x).
12. h(-5) = 10.
13. The equation is f(x) = 2x + 5.
14. The graph of y = f(x) will shift right 9 units.
What is the absolute value function?
The absolute value function is defined by the following piecewise rule, depending on the input of the function:
|x| = x, x ≥ 0.
|x| = -x, x < 0.
It measures the distance of a point x to the origin, hence, for example, |-2| = |2| = 0, and has vertex(turning point) at (0,0). The range is from the y-coordinate of the vertex to positive infinity.
In item 9, from the table, the turning point is at (2,0), hence:
Vertex (2, 0), Range: y
What is a linear function?
A linear function is modeled by the following rule:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, the change in y divided by the change in x.
b is the y-intercept, which is the the value of y when the function crosses the x-axis, that is, when x = 0.
For item 10, we have that:
The initial amount is of 100 patients, as 75 + 25 x 5 = 100 hence b = 100.
The reduction is of 5 patients in a week(25 patients reduction in 5 weeks = 5 a week), hence m = -5.
Then:
The equation is f(x) = -5x + 100.
For item 11, the slope is given by change in y divided by change in x, hence:
f(x): (1 - (1))/(3 -0) = 2/3 = 0.67.
g(x) = (4 - 2)/(5 - 0) = 0.4.
Hence:
The correct option is: The slope of f(x) is greater than the slope of g(x).
For item 12, we have that:
h(x) = -4x - 10.
Hence, when x = -5:
h(-5) = -4(-5) - 10 = 20 - 10 = 10.
Hence:
h(-5) = 10.
For item 13, we have that when x changes by 4, y changes by 8, hence the slope is:
m = 8/4 = 2.
When x = 1, y = 7, hence, considering the slope, when x = 0, y = 5, and:
The equation is f(x) = 2x + 5.
What is a translation?
A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s range(involving values of y) or in it’s domain(involving values of x). Examples are shift left, shift right, shift down, shift up, vertical stretching, horizontal stretching, vertical compression, horizontal compression, reflection over the x-axis, reflection over the y-axis, and rotations by a determined amount(in degrees). All these translations have predetermined rules that we apply to the figures.
For this problem, the rule is given by:
y = f(x - 9).
The translation is x -> x - 9, hence:
The graph of y = f(x) will shift right 9 units.