Marty and Ethan both wrote a function, but in different ways.
Marty
[tex]$v+3=\frac{1}{3}(x+9)$[/tex]
Ethan
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline-4 & 9.2 \\
\hline-2 & 9.6 \\
\hline 0 & 10 \\
\hline 2 & 10.4 \\
\hline
\end{tabular}
Whose function has the larger slope?
A. Marty's with a slope of [tex]$2 / 3$[/tex]
B. Ethan's with a slope of [tex]$2 / 5$[/tex]
C. Marty's with a slope of [tex]$1 / 3$[/tex]
D. Ethan's with a slope of [tex]$1 / 5$[/tex]
Answer (2)
Find the slope of Marty's function by rewriting the equation v + 3 = 3 1 ( x + 9 ) in slope-intercept form, resulting in a slope of 3 1 .
Find the slope of Ethan's function using two points from the table, such as (-4, 9.2) and (-2, 9.6), resulting in a slope of 5 1 .
Compare the slopes 3 1 and 5 1 .
Determine that Marty's function has the larger slope: Marty’s with a slope of 1/3 .
Explanation
Problem Analysis We are given two functions, one in equation form and one in a table. We need to find the slope of each function and compare them to determine which is larger.
Finding Marty's Slope First, let's find the slope of Marty's function. The equation is given as v + 3 = 3 1 ( x + 9 ) . To find the slope, we need to rewrite this equation in slope-intercept form, which is v = m x + b , where m is the slope and b is the y-intercept.
Distribute the 3 1 on the right side: v + 3 = 3 1 x + 3 1 ( 9 ) v + 3 = 3 1 x + 3
Subtract 3 from both sides: v = 3 1 x + 3 − 3 v = 3 1 x
So, the slope of Marty's function is 3 1 .
Finding Ethan's Slope Next, let's find the slope of Ethan's function. We can use any two points from the table. Let's use the points (-4, 9.2) and (-2, 9.6). The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1
Plugging in the coordinates: m = − 2 − ( − 4 ) 9.6 − 9.2 m = 2 0.4 m = 0.2
So, the slope of Ethan's function is 0.2 , which is equal to 5 1 .
Comparing the Slopes Now, we compare the slopes of the two functions: Marty's slope: 3 1 Ethan's slope: 5 1
To compare them easily, we can convert them to decimals: 3 1 ≈ 0.333 5 1 = 0.2
Since 0.2"> 0.333 > 0.2 , Marty's function has a larger slope.
Final Answer Therefore, Marty's function has the larger slope, which is 3 1 .
Examples
Understanding slopes is crucial in many real-world applications. For instance, in construction, the slope of a ramp determines its steepness and accessibility. In economics, the slope of a supply or demand curve indicates how responsive the quantity is to changes in price. In physics, the slope of a velocity-time graph represents acceleration. Knowing how to calculate and compare slopes allows us to analyze and predict changes in various scenarios.
Marty's function has a slope of 3 1 , while Ethan's has a slope of 5 1 . Comparing these values shows that Marty's slope is larger. Thus, the answer is Marty's with a slope of 3 1 .
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