Domain and Range from Graphs: Reading Graphs, Vertical Line Test, and Point-Slope Form

The goal: read domain and range from a graph. The CRA (course assessment) will often ask you to identify these.
Plan for today: work an example together to understand inputs, outputs, and how to express them.

Domain: the set of all inputs x for which the graph has a corresponding output. It is read left to right on the x-axis.
Range: the set of all outputs y produced by the graph. It is read bottom to top on the y-axis.
In practice, we express domain and range using interval notation so a computer or software can understand it easily.
Notation basics:
- Domain describes the x-values: left to right.
- Range describes the y-values: bottom to top.
Example discussion from the transcript (illustrative):
- If the graph starts at x = 0 and ends at x = 5, the domain is D = [0,5].
- If the dot ended at x = 6 (instead of 5), the domain would be D = [0,6].
- If you start at x = 1 instead of 0, and the plotted y-values go from y = 1 to y = 5, the range would be R = [1,5] (example depending on the graph).
Important caveat: the actual domain/range depend on the plotted graph; if the line or curve continues beyond the shown portion, the domain/range should reflect that extension only if the graph actually extends.

Inputs are the x-values you can plug into the graph; they correspond to the horizontal axis.
Outputs are the y-values produced by the graph; they correspond to the vertical axis.
When asked to determine the domain and range, think:
- Domain = all x for which there is a corresponding y on the graph.
- Range = all y-values that occur as outputs on the graph.
The transcript emphasizes: reading graphs is a key skill; practice by tracing left-to-right for domain and bottom-to-top for range.

We express both domain and range as intervals, using brackets for endpoints that are included and parentheses for endpoints that are not included:
- Closed interval: [a,b] (both endpoints included)
- Open interval: (a,b) (neither endpoint included)
- Mixed: [a,b) or (a,b] depending on inclusion.
Examples:
- Domain from 0 to 5, inclusive: D = [0,5].
- If the domain goes from 0 to 6: D = [0,6].
- If the domain excludes 0 but includes 5: D = (0,5].
Quick mental map (from the lecture):
- Domain is read left-to-right (x-values).
- Range is read bottom-to-top (y-values).
When the line continues beyond the plotted endpoints, reflect that continuation in the interval; if not, keep endpoints as shown.

The vertical line test checks whether a graph represents a function.
How it works:
- Take a vertical line at some input value x and see how many points on the graph intersect that line.
- If there is more than one intersection (i.e., more than one output y for a single x), the relation is not a function.
In function terms:
- A function assigns to every input x exactly one output y. If a vertical line hits the graph in more than one point, that would violate the definition of a function.
Summary: The vertical line test demonstrates the uniqueness of outputs for each input.

A linear function is a line with slope m and y-intercept b.
Two common representations:
- Slope-intercept form: y = mx + b
- Point-slope form: y - y1 = m(x - x1) where \( (x1, y1) \) is any point on the line.
Why use point-slope form? It can be easier because you can plug in a known point on the line and the slope, and you’re effectively done after a single step.
Converting to slope-intercept form from point-slope form:
- Distribute the slope and solve for y to obtain y = mx + b with b = y1 - m x1.
Consistency: If you pick a different point \((x1', y1')\) on the same line, you should obtain the same slope m and the same intercept b when simplified.

Given slope m = rac{9}{4} and a point on the line \( (x1, y1) = (1, 3) \).
Start with the point-slope form:
- y - y1 = m(x - x1)
- Substituting: y - 3 = rac{9}{4}(x - 1)
Solve for y to get slope-intercept form:
- Distribute: y - 3 = rac{9}{4}x - rac{9}{4}
- Add 3 (which is rac{12}{4}) to both sides:
- y = rac{9}{4}x - rac{9}{4} + 3 = rac{9}{4}x - rac{9}{4} + rac{12}{4}
- Combine constants: y = rac{9}{4}x + rac{3}{4}
Therefore, the intercept is b = rac{3}{4} in this example.
Verification with another point on the same line: using a different point \( (x1', y1') \) on the line would yield the same intercept after simplification, demonstrating consistency.
Quick takeaway: \(b = y1 - m x1\) for any chosen point on the line; this formula yields the same b for any valid point on the line.

From point-slope to slope-intercept: start with y - y1 = m(x - x1), then rewrite as y = mx + (y1 - m x1).
Steps can be performed in any order, but distributing and combining constants is the standard route to the slope-intercept form.
The key intuition: point-slope form ties directly to a known point; slope-intercept form makes the slope and intercept explicit for quick graphing and interpretation.

When using online homework systems (e.g., WebWorks), students sometimes encounter formatting or input acceptance issues. Double-check that you’re entering exact forms (fractions vs decimals, correct signs, etc.).
Help resources mentioned in the transcript:
- Canvas (course portal)
- Instructor office hours
- Mathematics Resource Center (MRC)
- Teaching Assistants (TAs)
Real-world relevance: Understanding domain, range, and linear representations is foundational for modeling real-world problems with functions and for converting between different but equivalent representations of a model.
Social context from the transcript (for study planning): opportunities to study together, join office-hours sessions, and utilize campus resources.

Domain and range concepts (in words):
- Domain = set of all inputs x; Range = set of all outputs y.
Interval notation conventions (for endpoints):
- Closed interval: [a,b]
- Open interval: (a,b)
- Mixed: [a,b) or (a,b]
Vertical line test (function criterion):
- A relation is a function if and only if every vertical line intersects the graph at most once.
Linear forms:
- Slope-intercept: y = mx + b
- Point-slope: y - y1 = m(x - x1)
- Relationship between intercept and a point: b = y1 - m x1
Worked example (with numbers):
- Point-slope: y - 3 = rac{9}{4}(x - 1)
- Slope-intercept: y = rac{9}{4}x + rac{3}{4}
Quick practice prompt: If a graph represents a line with slope m = rac{9}{4} passing through \( (1,3) \), the y-intercept is b = 3 - rac{9}{4}(1) = rac{3}{4} and the equation is y = rac{9}{4}x + rac{3}{4}.
Help and study tips: use Canvas and office hours; practice with interval notation; check functional status with vertical line test; practice converting between point-slope and slope-intercept forms to build flexibility in solving problems.