When you are asked which rule describes the function whose graph is shown, you are essentially being asked to reverse-engineer a mathematical relationship from a visual picture. Still, a graph is nothing more than a collection of ordered pairs plotted on a coordinate plane, and every point on that graph must satisfy the function’s rule. Whether the graph is a straight line, a curved parabola, or a rapidly climbing curve, there is a specific algebraic expression—often written in function notation like f(x)—that generates those points. Learning how to identify that rule is a foundational skill in algebra, and it bridges the gap between visual intuition and symbolic mathematics Which is the point..
What Does “Which Rule Describes the Function Whose Graph Is Shown” Mean?
In mathematics, a function is a rule that assigns exactly one output to each input. Still, when you see a graph, you are looking at a visual representation of that rule. The question “which rule describes the function whose graph is shown” requires you to examine key features of the graph—such as intercepts, slope, vertex, direction, and shape—and translate those observations into an equation.
The official docs gloss over this. That's a mistake The details matter here..
To give you an idea, if a graph is a straight line passing through the points (0, 2) and (2, 6), you can determine that the slope is 2 and the y-intercept is 2, which leads to the rule f(x) = 2x + 2. If the graph is a U-shaped curve with its lowest point at (1, -3) and passing through (0, -2), you might conclude the rule is quadratic, such as f(x) = (x - 1)² - 3 And that's really what it comes down to..
The goal is to move from “what I see” to “what the equation says.”
Common Function Types and Their Graphs
Before you can name the rule, you need to recognize the family of functions the graph belongs to. Each family has a distinctive shape and behavior That's the part that actually makes a difference..
Linear Functions
A linear function graphs as a straight line. Its rule is typically written in slope-intercept form:
f(x) = mx + b
Here, m is the slope (the rate of change), and b is the y-intercept (where the line crosses the y-axis). If the line rises from left to right, the slope is positive. Think about it: if it falls, the slope is negative. Horizontal lines have a slope of zero, and vertical lines are not functions because they fail the vertical line test.
The official docs gloss over this. That's a mistake.
Quadratic Functions
A quadratic function graphs as a parabola—a U-shaped curve that opens either upward or downward. Its standard form is:
f(x) = ax² + bx + c
The vertex is the highest or lowest point on the parabola, and the axis of symmetry runs vertically through that vertex. Plus, if a is positive, the parabola opens upward and has a minimum point. If a is negative, it opens downward and has a maximum point Not complicated — just consistent..
Exponential Functions
An exponential function graphs as a curve that grows or decays rapidly. Its rule usually looks like:
f(x) = a · bˣ
where a is the initial value and b is the base. If b > 1, the function shows exponential growth. Because of that, if 0 < b < 1, it shows exponential decay. These graphs have a horizontal asymptote, often the x-axis (y = 0), which the curve approaches but never touches.
Absolute Value Functions
An absolute value function graphs as a V-shaped curve. Its basic rule is:
f(x) = |x|
Transformations can shift, stretch, or reflect this V-shape. The vertex of the graph is the point where the direction changes, and it is often the easiest feature to identify when writing the rule.
Step-by-Step Guide to Finding the Function Rule
You do not need to guess when identifying a function rule from a graph. Follow a consistent process to increase your accuracy.
Step 1: Identify the function family.
Look at the overall shape. Is it a line? A parabola? A J-shaped curve? A V-shape? This narrows down the possible forms of the rule.
Step 2: Locate easy-to-read points.
Find the y-intercept, x-intercepts, vertex, or any point with clear integer coordinates. These points will help you solve for unknown constants in the equation.
Step 3: Determine key parameters.
For a line, calculate the slope using two points: (y₂ - y₁) / (x₂ - x₁). For a parabola, identify the vertex (h, k) and use the vertex form f(x) = a(x - h)² + k. For an exponential function, identify the initial value and whether the graph is growing or decaying.
Step 4: Substitute and solve.
Plug the coordinates of a known point into the general equation to solve for any remaining unknowns, such as a, b, or c Worth keeping that in mind..
Step 5: Verify with another point.
Choose a second point on the graph and substitute it into your proposed rule. If the equation holds true, you have likely found the correct function.
Worked Examples
Example 1: A Linear Graph
Imagine a graph that is a straight line passing through (0, 3) and (2, 7).
- The y-intercept is clearly 3, so b = 3.
- The slope is (7 - 3) / (2 - 0) = 4 / 2 = 2, so m = 2.
- The rule is f(x) = 2x + 3.
Check with another point, such as (4, 11): f(4) = 2(4) + 3 = 11. It matches.
Example 2: A Quadratic Graph
Suppose a parabola has its
Example 2: A Quadratic Graph
Suppose a parabola has its vertex at ((-1,,4)) and passes through the point ((0,,3)) It's one of those things that adds up. Practical, not theoretical..
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Choose the vertex form – because the vertex is known, the equation can be written as
[ f(x)=a,(x+1)^{2}+4 . ] -
Use the extra point to find (a) – substitute ((0,3)) into the formula:
[ 3 = a,(0+1)^{2}+4 \quad\Longrightarrow\quad 3 = a\cdot 1 + 4 . ]
Solving for (a) gives (a = -1). -
Write the complete rule – plug the value back:
[ f(x)= -,(x+1)^{2}+4 . ] -
Check with another clear point – the graph also shows ((-2,,3)).
[ f(-2)= -,(-2+1)^{2}+4 = -,(-1)^{2}+4 = -1+4 = 3, ]
which matches the observed coordinate, confirming the rule.
Example 3: An Exponential Growth Curve
A curve rises sharply from the point ((0,2)) and approaches the line (y=0) as (x) becomes very negative. Two easy points are ((0,2)) and ((3,16)) Small thing, real impact..
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The initial value (a) is the y‑intercept, so (a=2).
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Because the function is increasing, the base (b) must be greater than 1.
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Insert the two points into (f(x)=a,b^{x}):
[ \begin{cases} 2 = 2,b^{0} &\text{(trivially true)}\[4pt] 16 = 2,b^{3} \end{cases} \quad\Longrightarrow\quad b^{3}=8 \quad\Longrightarrow\quad b=2 . ]
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The rule is therefore (f(x)=2\cdot 2^{x}=2^{,x+1}) Not complicated — just consistent..
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Verification with a third point, say ((-1,1)):
[ f(-1)=2^{,(-1)+1}=2^{0}=1, ]
which matches the graph But it adds up..
Example 4: An Absolute‑Value V‑Shape
The V‑shaped graph has its vertex at ((‑2,,5)) and passes through ((0,,7)).
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Start with the translated absolute‑value form:
[ f(x)=c;|,x+2,|+5 . ]
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Use the point ((0,7)):
[ 7 = c;|,0+2,|+5 ;\Longrightarrow; 7 = c\cdot 2 +5 ;\Longrightarrow; c = 1 . ]
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The final equation is
[ f(x)=|,x+2,|+5 . ]
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Test with another visible point, ((‑4,,3)):
[ f(-4)=|, -4+2 ,|+5 = | -2 |+5 = 2+5 = 7, ]
which does not agree with the observed y‑value, indicating a mis‑read coordinate; re‑examine the graph or choose different points for a consistent rule.
Conclusion
Identifying a function rule from a graph becomes a reliable routine when the analyst first recognizes the family of the curve, then selects convenient, accurately read points, and finally solves for the unknown constants using the appropriate algebraic form. Whether the shape is linear, quadratic, exponential, or absolute‑value, the five‑step process—classify, locate points, determine parameters, substitute and solve, verify—provides a clear pathway from visual information to a precise algebraic expression. By consistently applying these steps, students and practitioners alike can translate any well‑drawn graph into its exact mathematical rule.
