Graph and Intercepts

Graphing and Intercepts Notes

• Goal: Understand graphing lines using slope-intercept form and intercepts, and how to move between forms (slope-intercept and standard form).

Key Concepts

• A line can be written in standard form: ax + by = c and in slope-intercept form: y = mx + b.
• m represents the slope (rise over run). The fraction form is m = \frac{rise}{run}.
• b represents the y-intercept: the point where the line crosses the y-axis, at x = 0, i.e., the point (0, b).
• Any point on the line is an ordered pair $(x, y)$. The line contains infinitely many such points.
• Intercepts are specific points where the line crosses the axes:
  • y-intercept: where the line crosses the y-axis (set x = 0).
  • x-intercept: where the line crosses the x-axis (set y = 0).
• To graph a line from slope-intercept form, you start at the y-intercept and then move by the slope (rise over run).

Forms of a Line

• Standard form: ax + by = c
• Slope-intercept form: y = mx + b
• Intercept form (conceptual): starting point on the axis plus a constant rate of change.

Slope-Intercept Form: Interpreting m and b

• s: y = mx + b
• m is the slope: rise over run. If m > 0, line goes up as you move right; if m < 0, line goes down.
• The run (horizontal movement) is the denominator of the slope; the rise (vertical movement) is the numerator.
• b is the y-intercept: the starting point on the y-axis when x = 0, i.e., the point (0, b).
• Example interpretation steps:
  • For y = 3x, m = 3 (or \frac{3}{1}), b = 0, intercept at $(0,0)$, slope up 3 for every 1 unit right.
  • For y = -2x, m = -2 (or \frac{-2}{1}), b = 0, intercept at $(0,0)$, slope down 2 for every 1 unit right.
  • For y = 2x + 1, m = 2, b = 1, intercept at $(0, 1)$, slope up 2 for every 1 unit right.

Checking If a Point Is On the Line (Plug-and-Check)

• For a line given by ax + by = c, a point (x0, y0) lies on the line iff a x0 + b y0 = c.
• Example (consistent equation): If the line is 2x + y = 5, then
  • For (1, 3): 2(1) + 3 = 5 \Rightarrow (1,3) ext{ is on the line}.
  • For (4, -2) : 2(4) + (-2) = 8 - 2 = 6
    eq 5 \Rightarrow (4,-2) ext{ is not on the line}.

Finding a Point on the Line (Given y = mx + b)

• You can choose any x value, then compute y = m x + b, obtaining a point $(x, y)$ on the line.
• Example: If y = 2x + 1 and x = 3, then y = 2(3) + 1 = 7, so the point (3, 7) lies on the line.
• Alternatively, you can choose any y and solve for x: for y = mx + b, x = \frac{y - b}{m} (when m \neq 0).

Graphing from Slope-Intercept Form

• For a line in the form y = mx + b:
  • Identify the slope m (as a fraction if needed, e.g., m = \frac{p}{q}).
  • Identify the y-intercept b; start at the point (0, b) on the graph.
  • Use the slope to plot the second point: move up (if m>0) or down (if m<0) by the numerator and to the right by the denominator: rise by the numerator, run by the denominator.
• Examples:
  • y = 3x + 0: start at (0,0); slope \frac{3}{1} → up 3, right 1.
  • y = -2x + 0: start at (0,0); slope \frac{-2}{1} → down 2, right 1.
  • y = 2x + 1: start at (0,1); slope \frac{2}{1} → up 2, right 1.
  • y = -3x + 4: start at (0,4); slope \frac{-3}{1} → down 3, right 1.
• Fractions in slope: e.g., m = \frac{1}{2}, starting at (0, b) , move up 1, right 2 each step.

Special Cases: Vertical and Horizontal Lines

• Vertical lines: x = a
  • Slope is undefined (division by zero).
  • No y-intercept (the line never crosses the y-axis for any finite y).
  • All points have the same x-coordinate: (a, y) for any y.
• Horizontal lines: y = b
  • Slope is 0 (rise 0 for any run).
  • y-intercept is at (0, b) ; x can be any value.
  • All points have the same y-coordinate: (x, b) for any x.

From Standard Form to Slope-Intercept Form

• Given ax + by = c with b \neq 0:
  • Solve for y: by = -ax + c \Rightarrow y = -\frac{a}{b}x + \frac{c}{b}
  • Slope m = -\frac{a}{b}; y-intercept b_{int} = \frac{c}{b} (the point (0, \frac{c}{b}) ).
• If you want the x-intercept from standard form, set y = 0: ax = c \Rightarrow x = \frac{c}{a} (provided a \neq 0).
• If you want the y-intercept from standard form, set x = 0: by = c \Rightarrow y = \frac{c}{b} (provided b \neq 0).

Graphing from Intercepts

• You can graph a line using its intercepts:
  • y-intercept: (0, b) where b = y_{int}.
  • x-intercept: (\frac{c}{a}, 0) from the standard form expression, or from the equation by setting y = 0.
• With two intercept points, you can draw the line through them.
• Example practice (intercepts from a standard form line):
  • For 4x - 5y = 10:
  • y-intercept: set x = 0
    ightarrow -5y = 10
    ightarrow y = -2, so (0, -2) .
  • x-intercept: set y = 0
    ightarrow 4x = 10
    ightarrow x = \frac{5}{2} , so (\frac{5}{2}, 0) .
• If you have intercepts directly, you can plot them and draw the line through them, e.g., intercepts (2, 0) and (0, 4) define the line passing through those two points.

Worked Practice Examples (from the transcript-style reasoning)

• Example 1: Determine intercepts for line given in slope-intercept form y = 2x + 3
  • y-intercept: b = 3 \,\Rightarrow\, (0,3)
  • x-intercept: set y = 0: 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}
  • x-intercept: ( -\tfrac{3}{2}, 0)
• Example 2: Convert a standard form to slope-intercept and read intercepts
  • Given: 4x - 5y = 10
  • Solve for y: -5y = -4x + 10 \Rightarrow y = \frac{4}{5}x - 2
  • Slope: m = \frac{4}{5}; y-intercept: b = -2, so intercept at $(0, -2)$.
  • x-intercept from this form: set y = 0: 0 = \frac{4}{5}x - 2 \Rightarrow x = \frac{10}{4} = \frac{5}{2}, so x-intercept at $(\frac{5}{2}, 0)$.
• Example 3: Vertical line
  • Given: x = 3, slope undefined, no y-intercept, all points with x = 3.
• Example 4: Horizontal line
  • Given: y = -2, slope 0, intercept at $(0, -2)$, any x-value is allowed.

Intercept-Only Graphs (Two Intercepts) and Real-World Relevance

• In finance or physics contexts, intercepts can represent starting points (e.g., initial cost or initial position) and break-even points where profit/loss crosses zero.
• Intercepts help identify critical thresholds where the behavior of a system changes (e.g., break-even in a profit model, touchdown point in a projectile’s height vs. time when projected onto axes).

Quick Tips and Common Pitfalls

• Always identify m and b when you have a line in the slope-intercept form; do not confuse the coefficient of x (which is m) with the y-intercept value.
• When m is a fraction, keep the rise over run interpretation consistent (numerator = rise, denominator = run).
• If you are given standard form, you will often need to solve for y to read off m and b, unless you directly use intercepts instead.
• Reduce fractions for slopes when reporting them (e.g., prefer \frac{1}{2} to \frac{2}{4}).
• For vertical lines, remember the slope is undefined and there is no y-intercept.
• For horizontal lines, the slope is 0 and the y-intercept is the constant value on the line.

Summary and Takeaways

• The slope-intercept form y = mx + b makes it easy to read the slope and y-intercept directly and to graph the line by starting at $(0, b)$ and applying the rise/run determined by m = \frac{rise}{run}.
• The x- and y-intercepts give two concrete points on the line and are convenient for graphing, especially from standard form.
• Converting between standard form and slope-intercept form allows you to quickly find both the slope and intercepts, and to graph lines efficiently.
• Always verify whether a given point lies on a line by substitution, and remember special cases (vertical/horizontal) as you practice graphing.