Intercepts, Axes, and Windowing in Linear Graphs

Intercepts and Axes

The graph uses two axes: the x-axis (horizontal) and the y-axis (vertical). They cross at the origin (0,0).
The y-axis counts up and down; the x-axis counts left and right.
The y-intercept is the point where the graph crosses the y-axis (i.e., the value of y when x = 0).
The x-intercept is the point where the graph crosses the x-axis (i.e., the value of x when y = 0).
In a linear equation of the form y = mx + b:
- The y-intercept is the point (0, b) .
- The x-intercept is the point ( -\frac{b}{m}, 0 ) provided that m \neq 0.
The origin (0,0) is a reference point where the axes meet; intercepts tell you where the line meets the axes specifically.

Algebraic substitution method:
- Y-intercept: plug in x = 0 into the equation, yielding the y-coordinate of the intercept (often called the constant term, or simply b for lines in slope-intercept form).
- X-intercept: set y = 0 and solve for x: 0 = mx + b \Rightarrow x = -\frac{b}{m} (if m \neq 0).
- The intercepts are the points: (0, b) and ( -\frac{b}{m}, 0 ) .
Graphing calculator / graph tool method:
- Enter the equation into the graph function.
- Graph the line and read off where it intersects the axes:
- Y-intercept: the point where the graph crosses the y-axis (the value of y when x = 0).
- X-intercept: the point where the graph crosses the x-axis (the value of x when y = 0).
- If the intercepts are not visible, adjust the window (the range of x and y values shown) so the axes and intercepts are inside view.
Homework setup note from transcript:
- Some calculators have an "Equation Editor" or input field for entering the function; you graph from there.
- You may align your input to reflect the zero-zero reference used in the homework, i.e., consider the position where x = 0 as the starting reference for reading intercepts.

Suppose the line is y = -3x + 5
- Y-intercept: set x = 0 → y = 5, so intercept is (0, 5) .
- X-intercept: set y = 0 → 0 = -3x + 5 \Rightarrow x = \frac{5}{3}, so intercept is (\frac{5}{3}, 0) .
Summary for this example: intercepts are (0,5) and (\tfrac{5}{3}, 0) .

Window selection matters: you need a view that includes both intercepts and the axes so you can clearly see where the line crosses them.
If you start with a very tight window (e.g., only 0 to 60 on the x-axis), you might not see where the x-axis crossing occurs or even the axes themselves.
To ensure visibility of intercepts, you may expand the window to include negative and larger positive ranges.
The transcript’s practical example: to visualize a balance/payments scenario, you might set a window from roughly -5000 to 80000 on the x-range so the axes and intercepts are clearly visible on screen.
Interpreting the graph in real-world terms:
- The y-axis value represents the balance (or another quantity) at time (x) units.
- The slope (m) shows the rate of change per unit of x; a negative slope indicates a decrease over time.
- The y-intercept (the point where the line meets the y-axis) represents the starting value (e.g., initial balance).
- The x-intercept represents the time when the balance hits zero (e.g., payoff time).
The transcript’s final observation: as a payment is made each month, the balance decreases, so the line slopes downward toward the x-axis; the end result is that the graph approaches the x-axis as payments accumulate.

Foundational principles:
- Slope-intercept form: y = mx + b, where m is the rate of change per unit of x and b is the initial value on the y-axis.
- Intercepts provide quick snapshots of starting values and break-even points with axes.
- A function’s zero corresponds to the x-intercept, i.e., the solution to f(x)=0 for the x-coordinate where the output is zero.
Real-world relevance:
- Linear models are used for simple financial plans, budgeting, and amortization where payments lead to decreasing balances.
- Intercepts help answer practical questions: "What is the initial balance?" and "After how many payments will the balance be paid off?".
Ethical/practical considerations:
- Intercept interpretation relies on accurate data and appropriate scaling; incorrect windowing or misleading scales can lead to misinterpretation of where intercepts really lie.
- In financial modeling, real-world factors (fees, interest compounding, etc.) may alter the linear model, so remember intercepts are tied to the model's assumptions.

If the line is horizontal (slope m = 0) and b \neq 0: there is no x-intercept (the line never crosses the x-axis).
If the line is horizontal with b = 0: every point on the x-axis is an intercept (infinitely many x-intercepts).
If the line is vertical (undefined slope): there is no y-intercept unless the vertical line passes through x = 0; in that special case, the line would cross the y-axis along all points with x = 0, which is not a standard y-intercept for a vertical line not at x=0.
For non-linear functions, intercepts are found similarly by evaluating at the axis values: y-intercept is the value at x=0; x-intercepts are solutions to f(x)=0.

Problem 1: For y = 2x + 4, find the intercepts.
- Y-intercept: plug in x=0 → y=4 → intercept (0,4) .
- X-intercept: set y=0 → 0=2x+4 \Rightarrow x=-2 → intercept (-2,0) .
Problem 2: For y = -3x + 0, find the intercepts.
- Y-intercept: (0,0) .
- X-intercept: set y=0 → 0=-3x \Rightarrow x=0 → intercept (0,0) (the line passes through the origin).
Problem 3: Graphically, for y = x - 7 with window x in [-10, 10] and y in [-20, 20], find intercepts.
- Y-intercept: (0, -7) .
- X-intercept: set y=0 → 0 = x - 7 \Rightarrow x = 7 → intercept (7, 0) .
Quick reflection: How does changing the slope m affect the x-intercept? Since x_{int} = -\frac{b}{m} for the line y = mx + b, increasing |m| (with b fixed) tends to bring the x-intercept closer to 0 (in magnitude).

The x-intercept and y-intercept are the natural anchors of a linear graph, telling you where the line crosses the axes.
Intercepts can be found algebraically via substitution or by reading off a graph, using a suitable window to view the intercepts clearly.
In real-world modeling, intercepts connect to initial values and payoff/zero-crossing times, and careful windowing is essential for accurate interpretation.