Question

Function 1 is defined by the equation y=4/5x+2

Function 2 is defined by the following table:
x y
0 1
1 1.5
2 2
3 2.5
Which function has a greater slope?

Answer 1

The slope of a linear function represents the rate at which the output variable (y) changes with respect to the input variable (x). The slope is often denoted by the letter "m" and can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are any two points on the line.

To find the slope of Function 1, we can compare the coefficient of x in its equation with the formula for slope. We see that the coefficient of x in y = (4/5)x + 2 is 4/5. Therefore, the slope of Function 1 is 4/5.

To find the slope of Function 2, we can choose any two points from the table and use the slope formula. Let's choose the points (0, 1) and (3, 2.5). Plugging in these values, we get:

m = (2.5 - 1) / (3 - 0) = 1.5 / 3 = 1/2

Therefore, the slope of Function 2 is 1/2.

Comparing the slopes, we can see that the slope of Function 1 (4/5) is greater than the slope of Function 2 (1/2). Therefore, Function 1 has a greater slope than Function 2.

Answer 2

Function 1 has the greatest slope.

Explanation:

Function 1

Function 1 is given in slope-intercept form, y = mx + b, where m is the slope (and b is the y-intercept).

Therefore, the slope of function 1 is ⁴/₅.

Function 2

To find the slope of function 2, use the slope formula.

`\boxed{\begin{minipage}{8cm}\underline{Slope Formula}\\\\Slope $(m)=(y_2-y_1)/(x_2-x_1)$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}`

Let (x₁, y₁) = (0, 1)

Let (x₂, y₂) = (2, 2)

Substitute the values into the formula:

`\implies m=(2-1)/(2-0)=(1)/(2)`

Therefore, the slope of function 2 is ¹/₂.

Greatest slope

To determine which function has the greatest slope, rewrite both slopes so that the denominator of the fractions are the same.

`\textsf{Slope of function 1}=(4)/(5)=(4 \cdot 2)/(5 \cdot 2)=(8)/(10)`

`\textsf{Slope of function 2}=(1)/(2)=(1 \cdot 5)/(2 \cdot 5)=(5)/(10)`

As 8 is greater than 5, the slope of function 1 is greater than the slope of function 2.