Free graphing resources · Original practice
Plot four original exponential functions from y=a·b^(x-h)+k. Compare growth, decay, horizontal shifts and vertical offsets using listed sample points.
Plot four original exponential functions from y=a·b^(x-h)+k. Compare growth, decay, horizontal shifts and vertical offsets using listed sample points.
Choose Letter or A4, then download a student sheet first. Keep the separate reference or answer file closed until you have attempted the case. Every worksheet, SVG and answer is free without signup.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Complete free PDF pack.
| Case | Focus |
|---|---|
| EX01 | Growth from one |
| EX02 | Growth with a shift |
| EX03 | Decay from four |
| EX04 | Shifted decay |
All examples use original fictional data and coordinate units. They are practice material, not measured classroom results, published questions or evidence about real people.
EX01: Growth from one
| Check | Worked result |
|---|---|
| 1 | y values: 1/4, 1/2, 1, 2, 4. |
| 2 | Horizontal shift h=0; vertical shift k=0. |
| 3 | Growth because base > 1. |
The first case shows the checking pattern. Individual answer files contain the completed graph and result for every case.
Begin by selecting the reference PDF for the specific exercise case you intend to use. Each case provides a finite window of x values, such as negative two through positive two for the first example. Use the supplied formula y equals a times b raised to the power of x minus h plus k to calculate each corresponding y value manually. For instance, in the first case, substitute x equals zero into the equation with base two to confirm the output is one. This step ensures you understand how the listed coordinates generate the plotted points before distributing the student version. The values are fictional practice material, so focus on the arithmetic consistency rather than real world measurement.
Examine the base value provided in the case definition to determine the function type. If the base is greater than one, such as the base two used in the first two cases, the function represents growth. If the base is between zero and one, like the fraction one half in the third and fourth cases, it represents decay. Check the horizontal shift and vertical offset parameters to see how the graph moves relative to the origin. For example, the fourth case includes a vertical offset of negative one. This classification helps students recognize the structural difference between increasing and decreasing exponential trends within the finite practice window provided in the diagrams.
After students complete the exercises using the student PDF, compare their results against the separate answer PDF. The answer key contains the exact calculated values derived from the listed coordinates and supplied parameters. Since physical printing tests have not been performed, verify that the plotted points align with the calculated table values on your screen before assuming any visual discrepancy is a student error. The answer key confirms the correct application of the horizontal shift and vertical offset. Use this comparison to identify if students are misinterpreting the finite window or incorrectly applying the base exponent rules in their manual calculations.
| Watch for | Check instead |
|---|---|
| Assuming the plotted points represent an unbounded real world process. | Recognize that the plotted points are a finite practice window generated from specific x values, not a claim about infinite real world behavior. |
| Confusing the base value for the vertical offset when identifying decay. | Identify decay by checking if the base is between zero and one, such as one half, rather than looking at the vertical offset parameter k. |
| Expecting physical print quality to match digital precision without verification. | Remember that physical printer checks have not been performed, so rely on the digital coordinates and supplied values for accuracy rather than visual print output. |
Choose the PDF that matches your paper. Print at Actual size / 100% and turn off Fit, Shrink and browser headers. SVG is editable but an editor can rescale it; the PDF is the simpler print route. Physical printer, classroom and curriculum tests have not been completed.
A base above 1 produces growth as x increases; a base between 0 and 1 produces decay.
In b^(x-h), h shifts the curve horizontally. Calculate x-h before applying the power.
No. It is a finite practice window for the supplied sample points.
Yes. The individual student, reference and answer files plus collection packs are free without signup. Credit Zumola under CC BY 4.0 when sharing or adapting.
A reliable way to practice is to start from a drawn curve and recover a possible equation of the form y = a·b^(x-h) + k, where b > 0 and b ≠ 1. The horizontal asymptote gives k directly: if the curve flattens toward y = 2, then k = 2. The horizontal shift h is the x-value where the exponential part equals 1, so the point (h, a + k) sits one vertical step of size a above the asymptote.
Choose two clear points to solve for a and b. Substitute both into y = a·b^(x-h) + k and divide one equation by the other to eliminate a. If the resulting ratio is positive, b is positive; if it is not, your chosen points or assumed h are inconsistent. This backward reading is a strong check because it forces every parameter to earn its place.
Most introductory examples use a positive a, so the curve rises above its asymptote on one side. A negative a reflects the curve below the asymptote. For y = -2·3^(x-1) + 4, the asymptote is y = 4, and because a = -2 the graph approaches 4 from below as x decreases. At x = 1 the exponential part is 1, giving y = 2, so the point (1, 2) is two units below the asymptote.
Check the direction with two more inputs. At x = 0, y = -2·3^(-1) + 4 = -2/3 + 4 = 10/3, about 3.33. At x = 2, y = -2·3 + 4 = -2. The values move from about 3.33 down through 2 to -2, confirming a decreasing curve that flattens toward 4 on the left and falls away on the right. The domain is all real x; the range is y < 4 because the curve never reaches the asymptote.
A blank coordinate grid is not an answer. The graph must show the asymptote, at least three labeled points, and arrowheads or clear end behavior. Check the y-intercept by substituting x = 0, not by guessing from the drawing. Check whether the curve crosses the asymptote: for a pure exponential with a ≠ 0, it should not. If your drawn curve crosses, either the asymptote is misplaced or a plotted point is wrong.
Endpoint questions matter when a task restricts the domain, for example x ≥ 0. Then the graph is a ray-like segment starting at a solid or open endpoint depending on the inequality. Substitute the endpoint value to place it, and mark open endpoints with a hollow circle. For unrestricted exponential graphs, the domain is all real numbers and there is no endpoint to close.
A curve has horizontal asymptote y = 1, passes through (0, 3) and (2, 7), and is assumed to have the form y = a·2^x + 1. Verify whether these points fit and state the range.
At x = 0: a·2^0 + 1 = a + 1 = 3, so a = 2. At x = 2: 2·2^2 + 1 = 8 + 1 = 9, not 7. The assumed base 2 does not fit both points. Using y = a·b^x + 1 with (0, 3) gives a = 2. Then 2·b^2 + 1 = 7 gives b^2 = 3, so b = √3. The rule is y = 2·(√3)^x + 1. Since a = 2 > 0, the range is y > 1, and the asymptote y = 1 is never reached.
Locate the point where the curve is exactly a units above or below the asymptote. Its x-coordinate is h in y = a·b^(x-h) + k.
Only in special shifted cases. Substitute x = 0 into the rule to check; do not assume the intercept from the asymptote alone.
The curve becomes a segment with an endpoint. Substitute the boundary x-value to place it, and use an open or closed circle according to the inequality.
