SeaWorld Slope Splash: Graphing Linear Functions in Point-Slope Form

Materials Needed

• Paper or graph paper
• Pencil and colored pencils
• Ruler
• Calculator, optional
• Timer, optional
• SeaWorld-themed practice problems below

Learning Target

I can graph linear functions given in point-slope form, with or without converting the equation, and identify and interpret key features of the graph.

Objectives

By the end of this 15-minute lesson, the learner will be able to:

• Identify the slope and a point from a point-slope equation.
• Graph a line directly from point-slope form.
• Convert point-slope form to slope-intercept form when helpful.
• Identify and interpret the slope, a point, the y-intercept, and the x-intercept when appropriate.
• Rearrange a simple linear equation to isolate a quantity of interest.

Success Criteria

I am successful if I can:

• Correctly locate the given point.
• Use the slope as “rise over run.”
• Draw a straight line through the points.
• Explain what the slope means in the SeaWorld situation.
• Show accurate algebra when isolating a variable.

Introduction and Bell Work — Reinforce Prior Knowledge (2 minutes)

Hook: Imagine that a SeaWorld trainer tracks the number of visitors entering a dolphin show. The graph forms a straight line because the same number of visitors enters every few minutes. How could you predict the number of visitors after a certain amount of time?

Bell Work: Complete the following without help. Then explain your thinking aloud or in writing.

1. In the equation y = 3x + 5, identify the slope and the y-intercept.
2. Starting at the point (2, 4), use a slope of 2 to find one more point on the line.
3. What does a positive slope tell you about a graph?

Quick check: Expected answers are slope 3, y-intercept 5, another point such as (3, 6), and the line increases from left to right.

Body: I Do, We Do, You Do

I Do: Model the Skill (4 minutes)

Introduce point-slope form:

y - y1 = m(x - x1)

Explain:

• m is the slope.
• (x1, y1) is a point on the line.
• The signs inside the parentheses may look different from the point’s coordinates. For example, y - 4 represents y1 = 4, while x + 2 represents x1 = -2.

SeaWorld example:

y - 4 = 2(x - 3)

Step 1: Identify the information.

• Slope: m = 2
• Point: (3, 4)

Step 2: Graph without converting.

1. Plot the point (3, 4).
2. Use the slope 2 = 2/1: move up 2 and right 1.
3. Plot another point, such as (4, 6).
4. Draw a straight line through the points.

Step 3: Convert if desired.

y - 4 = 2(x - 3)
y - 4 = 2x - 6
y = 2x - 2

Now the y-intercept is (0, -2). The slope is still 2.

Interpretation: If x represents time in minutes and y represents visitors counted in hundreds, the slope means that the number of visitors increases by 2 hundred visitors per minute.

Rearranging to isolate a quantity:

Suppose the equation is y = 2x - 2, and we want to find the time x when the visitor count is 10.

10 = 2x - 2
12 = 2x
x = 6

At 10 hundred visitors, the time is 6 minutes.

We Do: Solve Together (4 minutes)

Work through the problem with the learner. Ask the learner to explain each step before writing it.

Problem: A SeaWorld ride’s wait-time display is modeled by:

y + 3 = -1(x - 2)

1. Identify the slope and the point.
2. Graph the line directly from point-slope form.
3. Convert the equation to slope-intercept form.
4. Identify the y-intercept.
5. Explain what the negative slope means in this situation.

Guided solution:

• Slope: -1
• Point: (2, -3)
• From (2, -3), move down 1 and right 1 to get (3, -4).
• Convert: y + 3 = -x + 2, so y = -x - 1.
• y-intercept: (0, -1).
• The value decreases by 1 unit for every 1-unit increase in time.

Formative check: Ask, “Why is the point (2, -3) instead of (2, 3)?” The learner should explain that y + 3 is equivalent to y - (-3).

You Do: Independent SeaWorld Challenge (3 minutes)

Choose one challenge or complete both if time allows.

Challenge A: Dolphin Show Attendance

The number of visitors, y, in hundreds after x minutes is modeled by:

y - 5 = 3(x - 1)

1. Identify the slope and given point.
2. Graph the function directly from point-slope form.
3. Convert it to slope-intercept form.
4. Interpret the slope in context.

Challenge B: Find the Time

Use the same equation to determine when the number of visitors reaches 20 hundred.

Rearrange the equation to isolate x:

20 - 5 = 3(x - 1)

Expected answers:

• Slope: 3; point: (1, 5).
• Slope-intercept form: y = 3x + 2.
• The number of visitors increases by 3 hundred per minute.
• 20 - 5 = 3(x - 1), so 15 = 3(x - 1), 5 = x - 1, and x = 6 minutes.

Conclusion and Closure — Recap (2 minutes)

Ask the learner to complete this verbal or written exit ticket:

1. Point-slope form tells me the ______ and one ______ on the line.
2. To graph directly, I plot the point and use ______ as rise over run.
3. One reason to convert to slope-intercept form is to quickly identify the ______.
4. In one sentence, explain how the slope can describe a real SeaWorld situation.

Key takeaway: In y - y1 = m(x - x1), the slope is m and the given point is (x1, y1). You can graph by plotting the point and using the slope, or convert the equation to y = mx + b to find the y-intercept.

Assessment

• Formative: Bell work, questioning during modeling, point-and-slope identification, and guided graphing.
• Summative: The learner correctly graphs one point-slope equation, converts it when requested, identifies at least two key features, interprets the slope, and isolates a variable in a context problem.

Differentiation and Choice

• Support: Provide a labeled formula card, a slope triangle showing rise and run, and a table with columns for slope, point, next point, and intercept.
• Verbal support: Have the learner describe each graphing move aloud before drawing it.
• Digital option: Use a graphing calculator or online graphing tool to check the hand-drawn graph.
• Extension: Create a SeaWorld scenario with a point-slope equation, then state what the slope and intercept mean.
• Challenge: Rearrange y - y1 = m(x - x1) to isolate x symbolically: x = (y - y1)/m + x1, assuming m ≠ 0.