Question

math 1b hw: graphing lines using slope / slope-int. form
count the slope for each section of the roller coaster.

1. which section is the steepest, section 4, 5, or 6? explain / verify your answer using the slope.

error analysis: find and explain the error in the slope formula below.

1. (20,6), (-2,6)

$m=\frac{6-6}{-2-20}$
$=\frac{0}{-22}$

1. graph the line thru $(-4, -2)$,

with a slope of $\frac{3}{5}$

Explanation:

Step1: Calculate slope for Section 4

Pick two points on Section 4: e.g., $(0,8)$ and $(8,0)$. Slope formula is $m=\frac{y_2-y_1}{x_2-x_1}$.
$m_4=\frac{0-8}{8-0}=\frac{-8}{8}=-1$

Step2: Calculate slope for Section 5

Pick two points on Section 5: e.g., $(0,8)$ and $(6,0)$.
$m_5=\frac{0-8}{6-0}=\frac{-8}{6}=-\frac{4}{3}$

Step3: Calculate slope for Section 6

Pick two points on Section 6: e.g., $(0,0)$ and $(6,4)$.
$m_6=\frac{4-0}{6-0}=\frac{4}{6}=\frac{2}{3}$

Step4: Identify steepest section

Steepness is absolute value of slope: $|m_4|=1$, $|m_5|=\frac{4}{3}$, $|m_6|=\frac{2}{3}$.
$\frac{4}{3}>1>\frac{2}{3}$, so Section 5 is steepest.

Step5: Error analysis for slope formula

Given points $(20,6), (-2,6)$. The formula uses $\frac{6-6}{-2-20}$, but slope is $\frac{y_2-y_1}{x_2-x_1}$; the denominator should be $-2-20$, but the error is that the sign in calculation is misapplied, and the correct slope is $\frac{6-6}{-2-20}=\frac{0}{-22}=0$ (a horizontal line, slope 0, the work incorrectly mishandles the denominator sign but the core error is not recognizing horizontal line slope is 0, or miscalculating the denominator subtraction).

Step6: Graph line for (-4,-2) slope 3/5

Start at $(-4,-2)$. Slope $\frac{3}{5}$ means rise 3, run 5: from $(-4,-2)$, move right 5 to $x=1$, up 3 to $y=1$, so second point $(1,1)$. Draw line through $(-4,-2)$ and $(1,1)$.

Answer:

1. Slopes: Section 4: $-1$, Section 5: $-\frac{4}{3}$, Section 6: $\frac{2}{3}$
2. Steepest section: Section 5, since its absolute slope $\frac{4}{3}$ is the largest.
3. Error Analysis: The slope calculation has a sign error in the denominator, and the correct slope is $\frac{6-6}{-2-20}=0$ (horizontal lines have a slope of 0, the work incorrectly misrepresents the denominator subtraction result but the final slope should be 0).
4. Graph: Draw a line connecting $(-4,-2)$ and $(1,1)$ (and extending in both directions) on the coordinate grid.