Reading Functions from Graphs: Coordinates, Intercepts, and the Vertical Line Test

1) Reading Points (x, y) and Connecting Them to f(x)

When a graph represents a function f, every point on the graph has coordinates (x, y) that match the function relationship y = f(x). That means:

• The x-coordinate is the input value.
• The y-coordinate is the output value, so y = f(x).

How to read a point accurately

To read a point from a grid, move in two steps:

• Start at the origin (0,0).
• Move left/right to the correct x value (input).
• Then move up/down to the correct y value (output).

If the graph passes exactly through a grid intersection, you can read the coordinates directly. If it falls between grid lines, estimate using the scale (for example, each small square might represent 0.5 units).

Connecting a point to function language

If you see a point (3, 5) on the graph of f, you can write:

• f(3) = 5 (input 3 produces output 5)
• or (3, f(3)) = (3, 5)

If you see a point (-2, 1), then f(-2) = 1.

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Step-by-step: reading outputs from a graph

Suppose you want to find f(4) from a graph.

• Locate x = 4 on the x-axis.
• Move vertically until you hit the graph.
• From that intersection point, move horizontally to the y-axis to read the y-value.
• That y-value is f(4).

Be careful: if the graph is discrete (separate points), you can only read f(4) if there is actually a plotted point with x = 4.

Graph-reading practice (points and f(x))

Answer each as a statement about f.

• A point on the graph is (-1, 3). Write it using function notation.
• The graph shows a point at (2, -4). What is f(2)?
• You read that when x = 5, the graph’s y-value is about 1.5. Write f(5) approximately.

2) Intercepts: x-intercepts and y-intercepts as Inputs and Outputs

Intercepts are special points where the graph crosses an axis. They are useful because they summarize key inputs and outputs quickly.

y-intercept (output when input is 0)

The y-intercept is where the graph crosses the y-axis. On the y-axis, x = 0, so the y-intercept has the form (0, b).

• If the y-intercept is (0, b), then f(0) = b.
• Meaning: the output when the input is 0.

How to find it on a graph: look for where the graph touches or crosses the y-axis.

x-intercept(s) (inputs that produce output 0)

The x-intercepts are where the graph crosses the x-axis. On the x-axis, y = 0, so an x-intercept has the form (a, 0).

• If an x-intercept is (a, 0), then f(a) = 0.
• Meaning: the input(s) that make the output 0.

How to find them on a graph: look for where the graph touches or crosses the x-axis. A graph can have:

• no x-intercepts,
• one x-intercept,
• multiple x-intercepts.

Step-by-step: interpreting intercepts

• If you see an x-intercept at (-3, 0), write f(-3) = 0.
• If you see a y-intercept at (0, 2), write f(0) = 2.

Intercept questions (graph-based reasoning)

• A graph crosses the x-axis at x = -2 and x = 4. What two equations involving f can you write?
• A graph crosses the y-axis at y = -5. What is f(0)?
• A graph never touches the x-axis. What does that tell you about solutions to f(x) = 0?

3) The Vertical Line Test: Is the Graph a Function?

A graph represents a function if each input x is paired with exactly one output y. The vertical line test checks this directly on the graph.

How the vertical line test works

Imagine drawing vertical lines x = c for many different values of c.

• If any vertical line intersects the graph in more than one point, then the graph is not a function (one input gives multiple outputs).
• If every vertical line intersects the graph in at most one point, then the graph is a function.

Step-by-step: applying the test

• Pick an x-value where the graph looks “stacked” vertically (common trouble spots: loops, sideways curves, circles).
• Visualize a vertical line through that x-value.
• Count intersection points with the graph.
• If you ever count 2 or more intersections for the same x, it fails.

Examples to visualize (without needing equations)

• A U-shaped curve opening up or down typically passes the test (one y for each x).
• A circle fails the test (many x-values have two y-values).
• A sideways-opening curve often fails (some x-values hit the curve twice).
• A set of discrete points can still be a function if no two points share the same x-value with different y-values.

Function-or-not questions (use the vertical line test)

• A graph shows two points (2, 1) and (2, -3). Is it a function? Explain using the vertical line idea.
• A graph is a closed loop (oval shape). Would it pass the vertical line test? Why?
• A graph consists of points (-1, 4), (0, 4), (1, 4). Is it a function? What does the vertical line test say?
• A graph consists of points (-1, 2), (0, 2), (0, 5), (1, 2). Is it a function? Identify the x-value that causes failure.

4) Domain and Range from a Graph Using Interval Notation

From a graph, the domain is the set of x-values that appear on the graph, and the range is the set of y-values that appear on the graph. The main skill is describing these sets precisely, often with interval notation.

Reading domain from a graph (x-values covered)

To find the domain visually:

• Scan the graph from left to right.
• Identify the smallest x-value reached and the largest x-value reached.
• Check whether endpoints are included (solid point) or excluded (open circle).

Endpoint symbols:

• Solid (filled) point: the endpoint is included, use [ or ].
• Open circle: the endpoint is not included, use ( or ).

Reading range from a graph (y-values covered)

To find the range visually:

• Scan the graph from bottom to top.
• Identify the smallest y-value reached and the largest y-value reached.
• Again, decide whether endpoints are included or excluded.

Interval notation reminders (as used on graphs)

Discrete Graphs vs Continuous Curves (and How Domain/Range Descriptions Change)

Discrete graphs (separate points)

A discrete graph is made of individual plotted points (not connected). In this case:

• The domain is a list of specific x-values.
• The range is a list of specific y-values.
• Interval notation is usually not appropriate unless the graph truly includes every value in an interval (discrete graphs usually do not).

Example: Points at (-2, 1), (0, 3), (4, 3).

• Domain: {-2, 0, 4}
• Range: {1, 3}

Notice the range can repeat outputs (two different x-values can share the same y-value) and still be a function, as long as x-values do not repeat with different y-values.

Continuous curves (connected graph)

A continuous curve includes all points along the drawn path. In this case:

• The domain is often an interval (or a union of intervals).
• The range is often an interval (or a union of intervals).
• Interval notation is the standard way to describe domain and range.

Example: A curve starts at an open circle at x = -1, continues without breaks, and ends at a solid point at x = 5. Then the domain is (-1, 5].

Graphs with gaps or multiple pieces

Some graphs have breaks (holes, jumps, or separate pieces). Then domain and range may require multiple intervals.

Example description: One piece exists for x from -4 to -1 (including both), and another piece exists for x greater than 2 (but not including 2). Domain:

[-4, -1] ∪ (2, ∞)

Domain and range questions (interval notation and discrete vs continuous)

• A continuous graph runs from a solid point at x = -3 to an open circle at x = 2. Write the domain in interval notation.
• A continuous graph has lowest y-value at a solid point y = -1 and increases upward forever. Write the range.
• A discrete graph has points at x = -2, -1, 0, 3. How should you write the domain: as an interval or as a set? Write it correctly.
• A graph has a hole (open circle) at (1, 2) but otherwise continues through that x-region. How does that affect the domain? How does it affect the range?

Mixed Practice: Read Values, Intercepts, Function Test, Domain/Range

Set A: Discrete relation

A relation is shown by points: (-3, 2), (-1, 0), (0, 4), (2, 0), (2, 5).

• List the domain and range using set notation.
• Identify all x-intercepts and the y-intercept (if any).
• Does it represent a function? Use the vertical line test idea (focus on repeated x-values).

Set B: Continuous curve description

A curve starts at a solid point at (-2, 1), rises to a highest point at (1, 5), then decreases and ends at an open circle at (4, 2). The curve is drawn continuously between these endpoints.

• Estimate the domain in interval notation.
• Estimate the range in interval notation (pay attention to included/excluded endpoints).
• What is the y-intercept likely to represent on the graph? (State it as f(0) equals some value you would read from the graph.)
• Would the vertical line test pass for a single-peaked curve like this? Explain briefly.

Set C: Function or not (reasoning from shape)

• A graph looks like a circle centered at the origin. Decide if it is a function and justify using the vertical line test.
• A graph looks like two separate line segments: one from x=-5 to x=-2, and another from x=1 to x=3. Can it still be a function? What would you check?
• A graph includes a vertical segment at x=2 from y=-1 to y=4. Is it a function? What does the vertical line test say?