Reading a Graph as a Story
A function graph shows how outputs change as inputs move. To describe it in words, imagine moving along the horizontal axis from left to right. Track whether the graph rises, falls, or stays at the same height, and note where its behavior changes.
This description is useful even when a graph has no formula attached to it. It helps summarize the information shown and makes it possible to identify important inputs and outputs, such as a peak, a valley, or an intercept.
Keep the axes in mind as you read: the horizontal coordinate is the input \(x\), and the vertical coordinate is the output \(f(x)\). A point \((a,b)\) on the graph says that the input \(a\) has output \(b\), or \(f(a)=b\).
Increasing, Decreasing, and Constant Behavior
On a graph, these definitions have a direct visual meaning. As you move from left to right, an increasing section rises, a decreasing section falls, and a constant section is horizontal. The direction is determined by how the output changes, not by whether the graph lies above or below the horizontal axis.
For example, a graph below the horizontal axis can still be increasing if its outputs rise as the inputs increase. Likewise, a graph above the axis can be decreasing if its outputs fall as the inputs increase.
A graph may change behavior at a turning point or where a horizontal section begins or ends. Describe each interval separately. Two neighboring intervals may share a boundary input: this does not mean the function behaves in two conflicting ways there. It means the boundary separates sections with different behavior.
A Reliable Reading Process
Before describing a graph, identify its visible domain: the inputs for which points are actually shown. Pay attention to endpoint markers. A filled dot includes that point in the graph; an open circle does not. The graph’s domain and range can also help confirm whether the endpoints and extreme output values have been read correctly.
Record any endpoint value and note whether the endpoint is open or filled.
Mark where it rises, falls, or remains horizontal.
Record peaks, valleys, flat sections, and any disconnected pieces or gaps.
Use the axes and grid to identify values such as \(f(a)\), intercepts, and extreme outputs.
Trace it once more from left to right and confirm that every section and boundary is accounted for.
Worked Example: Three Different Behaviors
Worked Example: Three Different Behaviors
Suppose a graph consists of straight line segments joining the points \((-3,1)\), \((-1,5)\), \((2,5)\), and \((4,-1)\), in that order. Describe its behavior and find its \(y\)-intercept and \(x\)-intercept.
Step 1: Read the sections from left to right. From \(x=-3\) to \(x=-1\), the graph rises from output \(1\) to output \(5\), so it is increasing on \([-3,-1]\). From \(x=-1\) to \(x=2\), it stays at output \(5\), so it is constant on \([-1,2]\). From \(x=2\) to \(x=4\), it falls from output \(5\) to output \(-1\), so it is decreasing on \([2,4]\).
Step 2: Find the \(y\)-intercept. The \(y\)-intercept occurs at \(x=0\). Since \(0\) lies in the horizontal section from \(-1\) to \(2\), its output is \(5\). Thus \(f(0)=5\), and the \(y\)-intercept is \((0,5)\).
Step 3: Find the \(x\)-intercept. The graph crosses the horizontal axis on the segment from \((2,5)\) to \((4,-1)\). Its slope is \(\frac{-1-5}{4-2}=-3\). Starting at \((2,5)\), the line has output \(5-3(x-2)=11-3x\). Setting the output equal to zero gives \(11-3x=0\), so \(x=\frac{11}{3}\). The \(x\)-intercept is \((\frac{11}{3},0)\).
Check: Since \(2<\frac{11}{3}<4\), this intercept lies on the segment where the graph falls. Substituting \(x=\frac{11}{3}\) into \(11-3x\) gives \(11-11=0\), as required. The graph’s lowest output is \(-1\) at \(x=4\), and its highest output is \(5\) throughout the flat section.
Turning Behavior and Notable Values
A turning point is a point where the graph changes direction: for example, from increasing to decreasing, or from decreasing to increasing. A change to or from a constant section is also worth describing explicitly. A flat section can contain several inputs with the same output; it should not be reduced to a single turning point.
A local peak is an output that is higher than nearby outputs, while a local valley is lower than nearby outputs. The highest or lowest output on the entire displayed graph is called a global maximum or global minimum, respectively. A local peak need not be the global maximum, so compare it with the rest of the graph before making that claim.
Other useful features include endpoints, intercepts, and coordinates at which the graph changes behavior. When describing these, state both the input and the output. Saying “the graph reaches \(4\)” is less precise than saying “at input \(x=2\), the output is \(f(2)=4\).”
Worked Example: A Valley, a Flat Section, and a Falling End
Worked Example: A Valley, a Flat Section, and a Falling End
Suppose a graph consists of straight segments joining \((-4,0)\), \((-2,-2)\), \((1,2)\), \((3,2)\), and \((5,-2)\), in that order. Describe its behavior, identify its turning behavior, and find \(f(0)\) and the \(x\)-intercepts.
Step 1: Describe each section. From \(x=-4\) to \(x=-2\), the output falls from \(0\) to \(-2\), so the graph decreases on \([-4,-2]\). From \(x=-2\) to \(x=1\), it rises from \(-2\) to \(2\), so it increases on \([-2,1]\). From \(x=1\) to \(x=3\), it stays at \(2\), so it is constant on \([1,3]\). From \(x=3\) to \(x=5\), it falls from \(2\) to \(-2\), so it decreases on \([3,5]\).
Step 2: Describe the turns. At \((-2,-2)\), the graph changes from decreasing to increasing, so this is a valley. The graph rises to output \(2\), stays there from \(x=1\) through \(x=3\), and then falls. Thus it has a flat highest section on \([1,3]\), not just one peak point. The minimum output \(-2\) occurs at \(x=-2\) and \(x=5\).
Step 3: Find \(f(0)\). The input \(0\) lies on the segment from \((-2,-2)\) to \((1,2)\). The slope of this segment is \(\frac{2-(-2)}{1-(-2)}=\frac{4}{3}\). Starting from \((-2,-2)\), its output at \(x=0\) is \(-2+\frac{4}{3}(0-(-2))=-2+\frac{8}{3}=\frac{2}{3}\). Therefore \(f(0)=\frac{2}{3}\).
Step 4: Find the \(x\)-intercepts. The first segment meets the horizontal axis at its endpoint \((-4,0)\). On the segment from \((-2,-2)\) to \((1,2)\), the output is \(-2+\frac{4}{3}(x+2)\). Setting this to zero gives \(\frac{4}{3}(x+2)=2\), then \(x+2=\frac{3}{2}\), so \(x=-\frac{1}{2}\). On the last segment, the slope is \(\frac{-2-2}{5-3}=-2\), and its output is \(2-2(x-3)=8-2x\). Setting this to zero gives \(x=4\).
Check: The calculated intercepts, \(-4\), \(-\frac{1}{2}\), and \(4\), lie in the graph’s domain \([-4,5]\). Substitution gives outputs \(0\), \(-2+\frac{4}{3}(\frac{3}{2})=0\), and \(8-2(4)=0\). Each point is therefore on the horizontal axis.
Why the Left-to-Right Test Works
The descriptions “rises,” “falls,” and “stays level” are not guesses based on the graph’s overall shape. They follow directly from comparing outputs at ordered inputs.
This comparison also explains why the direction of travel matters. Moving from right to left reverses the order of the inputs, so it can make the visual description seem reversed. The standard convention is to describe behavior as \(x\) increases—that is, from left to right.
Worked Example: A Peak and Two Intercepts
Worked Example: A Peak and Two Intercepts
Suppose a graph consists of straight segments joining \((-3,-2)\), \((-1,2)\), \((1,2)\), and \((3,-2)\), in that order. Describe its behavior and find its \(x\)-intercepts and \(f(0)\).
Step 1: Describe the sections. From \(x=-3\) to \(x=-1\), the output rises from \(-2\) to \(2\), so the graph increases on \([-3,-1]\). From \(x=-1\) to \(x=1\), the output remains \(2\), so it is constant on \([-1,1]\). From \(x=1\) to \(x=3\), the output falls from \(2\) to \(-2\), so the graph decreases on \([1,3]\). The constant section is the graph’s highest section, and the endpoints have the lowest output.
Step 2: Find the first \(x\)-intercept. On the segment from \((-3,-2)\) to \((-1,2)\), the slope is \(\frac{2-(-2)}{-1-(-3)}=\frac{4}{2}=2\). Starting at \((-3,-2)\), the output is \(-2+2(x+3)=2x+4\). Set this equal to zero: \(2x+4=0\), so \(x=-2\).
Step 3: Find the second \(x\)-intercept. On the segment from \((1,2)\) to \((3,-2)\), the slope is \(\frac{-2-2}{3-1}=-2\). Starting at \((1,2)\), the output is \(2-2(x-1)=4-2x\). Setting \(4-2x=0\) gives \(x=2\). The intercepts are therefore \((-2,0)\) and \((2,0)\).
Step 4: Find \(f(0)\). Since \(0\) is in the constant section \([-1,1]\), its output is \(2\), so \(f(0)=2\). The \(y\)-intercept is \((0,2)\).
Check: Substituting the intercept inputs into the segment expressions gives \(2(-2)+4=0\) and \(4-2(2)=0\). Both intercepts lie in their respective segments. Also, \(0\in[-1,1]\), where the graph’s output is \(2\), confirming \(f(0)=2\).
Common Pitfall: Confusing Height with Direction
A common mistake is to call a section increasing because it lies above the horizontal axis, or decreasing because it lies below it. Those statements confuse the graph’s height with the way its height changes.
For example, the segment from \((-3,-2)\) to \((-1,2)\) in the previous example starts below the horizontal axis. A mistaken description might say, “It is decreasing because it is below the axis.” But as \(x\) moves from \(-3\) to \(-1\), the output changes from \(-2\) to \(2\), and \(-2<2\). The section is increasing. Conversely, the segment from \((1,2)\) to \((3,-2)\) starts above the axis but decreases because its output falls from \(2\) to \(-2\).
Another mistake is to report a turning point’s output without its input. A complete description includes both coordinates, such as “the graph reaches output \(2\) at every input from \(-1\) to \(1\).” If a graph has open circles, do not report them as included function values; check whether the point is filled before treating it as part of the graph.
Check Your Understanding
Use the graph information in each question. For questions involving segments, assume consecutive listed points are joined by straight line segments.
- A graph joins \((0,1)\), \((2,5)\), and \((4,1)\). On which intervals is it increasing and decreasing, and what is its highest output?
- A graph joins \((-2,3)\), \((1,3)\), and \((4,0)\). Describe its behavior from left to right. Is it constant on the whole domain?
- A graph is decreasing on \([1,3]\) and lies above the horizontal axis throughout that interval. Is it increasing or decreasing?
- A graph has a filled dot at \((2,4)\) and an open circle at \((5,4)\). Which of these points is included in the graph, and what output does the filled dot specify?
- A graph changes from decreasing to increasing at \((1,-3)\). What kind of turning behavior occurs there?
