Question

The table below represents a linear function f(x) and the equation represents a function g(x):

Part A: Write a sentence to compare the slope of the two functions and show the steps you used to determine the slope of f(x) and g(x). (6 points)

Part B: Which function has a greater y-intercept? Justify your answer. (4 points)

Answer 1

Answer:

Slope of f(x) is greater than slope of g(x)

g(x) has a greater y intercept

Step-by-step explanation:

Given

f(x) table

$g(x) = 2x + 6$

Solving (a):

First, we determine the slope of f(x).

From the table, we take any two corresponding values of x and f(x).

Represent f(x) with y

$(x_1,y_1) = (-1.-12)$

$(x_2,y_2) = (0.-6)$

The slope (m) is calculated as thus

$m = \frac{y_2 - y_1}{x_2 - x_1}$

$m = \frac{-6 - (-12)}{0 - (-1)}$

$m = \frac{-6 +12}{0 +1}$

$m = \frac{6}{1}$

$m = 6$

Represent this with m1

$m_1 = 6$

Calculating the slope of g(x).

The general form of an equation is $y = mx + b$

Where m represents the slope.

We have that:

$g(x) = 2x + 6$

By comparing  $g(x) = 2x + 6$ with $y = mx + b$

$m = 2$

Represent this with m2

$m_2 = 2$

Comparing both slope, we can say that:

f(x) has a greater than slope of g(x)

Another comparison is that:

Slope of f(x) is 3 times the slope of g(x)

Solving (b): Function with greater y intercept.

The general form of an equation is $y = mx + b$

Where b represents the y intercept.

First, we need to determine the equation of f(x) using:

$y - y_1 = m(x - x_1)$

Recall that, from the table of f(x):

$(x_1,y_1) = (-1.-12)$

$(x_2,y_2) = (0.-6)$

$m_1 = 6$

So:

$y - y_1 = m(x - x_1)$

$y - (-12) = 6(x - (-1))$

$y + 12 = 6(x +1)$

$y + 12 = 6x +6$

Solve for y

$y = 6x + 6 - 12$

$y = 6x - 6$

By comparing this with $y = mx + b$, the y intercept of f(x) is -6

For g(x), we have:

$g(x) = 2x + 6$

By comparing this with $y = mx + b$, the y intercept of g(x) is 6

Comparing the y intercepts of both functions, g(x) has a greater y intercept because $6 > -6$