Visualizing Problems
Equations and Functions
![Visualizing Problems
Equations and Functions
Turning real-world relationships into equations, tables, and graphs
Suggested image:
coordinate grid
with a line, points,
and simple equations]()
Learning Journey
![Learning Journey
1. What equations tell us
2. How functions connect inputs and outputs
3. Equation, table, and graph
4. How to read a graph
5. A taxi-cost model
6. Practice and review]()
Equations Show Equality
![Equations Show Equality
An equation says that two expressions
have the same value.
2x + 3 = 11
Subtract 3: 2x = 8
Divide by 2: x = 4
Check: 2(4) + 3 = 11
Suggested image: balance scale
showing equal expressions
on both sides]()
A Solution Is an Intersection
![A Solution Is an Intersection
Graph 1: y = 2x + 3
Graph 2: y = 11
Intersection: (4, 11)
Shared x-value: x = 4
The crossing point makes both
sides of the equation equal.
Suggested image: coordinate graph of
y = 2x + 3 and y = 11,
meeting at (4, 11)]()
Functions Connect Inputs and Outputs
![Functions Connect Inputs and Outputs
Input x
0
1
2
3
Rule
f(x) = 2x + 3
Output
3
5
7
9
A function assigns exactly one output to each input.]()
One Function, Three Views
![One Function, Three Views
Equation
y = 2x + 3
Shows the rule using
symbols and numbers.
Table
x y
0 3
1 5
2 7
3 9
Lists matching input-output pairs.
Suggested image:
line through the
points
(0,3), (1,5), (2,7),…
Reveals the visual pattern.]()
Reading a Graph
![Reading a Graph
Suggested image: labeled graph showing
x-intercept, y-intercept, slope,
intersection, maximum, and minimum
x-intercept: crosses the horizontal axis
y-intercept: crosses the vertical axis
Slope: shows how quickly y changes
Intersection: two graphs share a value
Maximum or minimum: highest or lowest point]()
Slope and Starting Value
![Slope and Starting Value
y = mx + b
m: slope or rate of change
b: y-intercept or starting value
Suggested image: line with a marked
y-intercept and a rise-over-run
slope triangle
For y = 2x + 3, the slope is 2 and the starting value is 3.]()
Modeling a Taxi Ride
![Modeling a Taxi Ride
A taxi ride has a $4 base fee
plus $2 for every mile.
C(d) = 2d + 4
d: distance in miles
C(d): total cost
2: cost per mile
4: starting fee
Suggested image: taxi-cost graph
starting at $4 and increasing
$2 for each mile]()
From a Situation to a Graph
![From a Situation to a Graph
1. Identify
Choose the input,
output, starting value,
and rate of change.
2. Write
Build an equation
that connects the
two quantities.
3. Calculate
Make a table of
input-output pairs
using the rule.
4. Plot
Place the points
on a coordinate grid
and describe the pattern.
Situation → Equation → Table → Graph → Interpretation]()
Try It Yourself
![Try It Yourself
A bike rental costs $6 to start and $3 for each hour.
Let h be the number of hours and R(h) be the total cost.
1. Write the function.
What is the starting value?
What is the hourly rate?
2. Complete a table.
Use h = 0, 1, 2, 3.
Find each total cost.
3. Sketch the graph.
Label both axes.
Explain the slope.]()
Key Ideas to Remember
![Key Ideas to Remember
Equations
Show that two expressions
have equal values.
Solving finds the value
that makes the statement true.
Functions
Connect every input
to exactly one output.
They describe how
quantities change together.
Graphs
Turn numbers and symbols
into visible patterns.
They reveal intercepts,
slope, and intersections.
Use all three views to understand and solve real-world problems.]()
