Reading a Secant Slope from Information
A graph or table often gives function values without giving a formula for the function. Even then, you can calculate the slope between two specified inputs. The key is to identify the two matching points, use the same endpoint order for both changes, and divide the change in function value by the change in input.
A graph-based calculation may be an estimate because plotted coordinates are sometimes read approximately. With a table of values, the arithmetic can be exact, but the result still describes only the selected pair of data points.
This is the average rate of change over the chosen interval, expressed as the slope of the line through the two function points. The important practical detail is that the same order of endpoints must be used in both differences.
Why the Formula Gives the Secant Slope
The secant line is the straight line through the two selected points. To see why its slope is the displayed quotient, write the equation of a line through \((x_1,f(x_1))\) using its slope \(m\):
The second point, \((x_2,f(x_2))\), lies on this line. Substituting its coordinates gives
Because the selected inputs are distinct, \(x_2-x_1\ne 0\), so dividing both sides by \(x_2-x_1\) gives
Thus the quotient is not an arbitrary calculation: it is the slope of the line joining the two function points. If the two function values are equal, the secant is horizontal and the slope is zero. If the output decreases as the input increases, the slope is negative.
Choosing and Reading the Two Points
For a graph, first locate the interval endpoints on the horizontal axis. Then read the corresponding function values on the vertical axis. A point on the graph has coordinates \((x,f(x))\): the input is the horizontal coordinate and the function value is the vertical coordinate. Do not use the vertical value as the denominator or the horizontal value as the numerator.
For a table, find the rows with the two chosen inputs and copy the matching function values. A row must be treated as a coordinate pair. For example, if a table says \(f(3)=-2\), that gives the point \((3,-2)\), not \((-2,3)\).
Choose the two inputs and make clear which point is first and which is second in your calculation.
Use graph coordinates or table entries to identify both points as \((x_1,f(x_1))\) and \((x_2,f(x_2))\).
Calculate \(f(x_2)-f(x_1)\) and \(x_2-x_1\), then divide the first change by the second.
State whether the slope is positive, negative, or zero; include units when the quantities have units; and verify that the quotient reproduces the vertical change.
Worked Examples
Worked Example: Estimating a Secant Slope from a Graph
A graph shows a function at two marked points. Reading the grid, the points are approximately \((1,2)\) and \((5,10)\). Estimate the secant slope over the interval from \(x=1\) to \(x=5\).
Use the first point as \((x_1,f(x_1))=(1,2)\) and the second as \((x_2,f(x_2))=(5,10)\). The output change is
The input change, in the same order, is
Therefore, the estimated secant slope is
Check: a slope of \(2\) over an input change of \(4\) predicts an output change of \(2\cdot4=8\), matching the change from \(2\) to \(10\). Since the coordinates were read from a graph, report this as an estimate; small reading errors could change the result.
Worked Example: Finding a Secant Slope from a Table
The table gives values of \(f\). Find the secant slope over the interval from \(x=-2\) to \(x=3\).
| \(x\) | \(f(x)\) |
|---|---|
| \(-2\) | \(7\) |
| \(0\) | \(3\) |
| \(3\) | \(-3\) |
| \(5\) | \(-7\) |
The selected rows give \((x_1,f(x_1))=(-2,7)\) and \((x_2,f(x_2))=(3,-3)\). First calculate the output change:
Next calculate the input change, maintaining the same order:
The secant slope is
Check: multiplying the slope by the input change gives \((-2)(5)=-10\), which is exactly the output change. The negative sign is consistent with the table: as \(x\) increases from \(-2\) to \(3\), \(f(x)\) decreases from \(7\) to \(-3\).
Worked Example: Keeping Subtraction Order Consistent
A table gives \(f(2)=3\) and \(f(6)=15\). Find the secant slope, but calculate from the point at \(x=6\) back to the point at \(x=2\).
Here the first point is \((x_1,f(x_1))=(6,15)\), and the second is \((x_2,f(x_2))=(2,3)\). Following that order, the output change is
The input change is
Thus
Check: \(3(-4)=-12\), as required for this direction of subtraction. Both changes are negative, so their quotient is positive. If the endpoints were instead used from \(2\) to \(6\), the changes would be \(15-3=12\) and \(6-2=4\), and the slope would still be \(12/4=3\). Reversing both changes leaves their quotient unchanged.
Units and Meaning
If the output and input have units, the slope has units of output per unit of input. For instance, if a table records distance in meters and time in seconds, then the secant slope has units of meters per second. This describes the average change in distance for each second across the selected time interval.
The sign gives useful information, but it does not by itself tell the whole story. A positive slope means the selected ending output is greater than the starting output when the input increases. A negative slope means the ending output is smaller. A zero slope means the two selected function values are equal. These claims concern the two selected points; the function could still rise or fall between them.
On a graph, estimate coordinates according to the axis scales, not merely the number of grid squares. If each vertical grid step represents \(2\) units, a point one vertical step higher has an output difference of \(2\), not \(1\). If a point does not lie exactly on a grid intersection, state that your coordinate and slope are approximate.
Common Mistake: Reversing Only One Difference
Suppose the points are \((2,3)\) and \((6,15)\). A common error is to calculate the output change from \(x=2\) to \(x=6\), but calculate the input change in reverse:
This is wrong because the numerator describes movement from the first point to the second, while the denominator describes movement from the second point back to the first. The correction is to use one consistent order:
Alternatively, reverse both differences: \((3-15)/(2-6)=(-12)/(-4)=3\). In either correct calculation, the check is that slope multiplied by the input change equals the output change in that same direction.
Check Your Understanding
For each question, identify the endpoints and keep the subtraction order consistent.
- A graph gives the points \((-1,4)\) and \((3,0)\). What is the secant slope from \(x=-1\) to \(x=3\)?
- A table gives \(g(2)=5\) and \(g(7)=20\). Find the secant slope from \(x=2\) to \(x=7\), and check it using the output change.
- For the points \((1,8)\) and \((5,2)\), calculate the slope from \(x=5\) back to \(x=1\). Why is the result unchanged from calculating in the opposite direction?
- A graph's vertical axis is marked in increments of \(3\). Two points differ by two vertical grid increments and four horizontal units. What is the secant slope if the output increases between the points?
- A quantity changes from \(12\) meters to \(3\) meters while time changes from \(2\) seconds to \(5\) seconds. What is the secant slope, including units, over this interval?
