Algebra homework
Graph a linear function from a table: check the pattern
Turn a value table into a reliable line by checking equal changes, plotting ordered pairs, and using AI feedback only after your own attempt.
Your worksheet gives a table with x values −2, 0, 2 and y values −3, 1, 5. It asks for a graph and an equation. The entries look simple, yet one reversed coordinate or uneven axis scale can produce a convincing wrong line. The useful question is not merely where to draw: does the table show a constant change, and do the plotted points agree with that change?
This guide works through the table, graph, slope, intercept, and checking steps before bringing in Lirno. If your school permits AI assistance, the app can help you inspect a complete scan and ask for a hint or check. You still need to make the mathematical decisions and compare any AI explanation with the actual worksheet. The examples also work with paper and a ruler alone.
1. Read each row as an ordered pair
A table of values connects each input x to an output y. Read one column or row at a time, depending on the worksheet layout. For the example, the pairs are (−2, −3), (0, 1), and (2, 5). The first coordinate always tells you the horizontal movement; the second tells you the vertical movement. Copying them as (y, x) is an easy way to create a graph that looks orderly but represents a different relationship.
Circle the headings and mark the direction of the x and y axes before plotting. If the table has units, keep them attached: minutes may be x and distance may be y. An x value of zero can describe a starting amount, whereas a y value of zero identifies where a line reaches the horizontal axis. The distinction matters in a word problem, where the coordinates tell a story as well as giving locations on paper.
2. Test for equal rates of change
A linear function has a constant rate of change. In our table, x rises by 2 from −2 to 0 and again by 2 from 0 to 2. Over those same intervals, y rises by 4 and 4. The ratio of vertical change to horizontal change is 4 divided by 2, or 2, for both intervals. That repeating ratio supports a straight-line graph rather than a curve.
Do not require the raw y differences to match when the x gaps differ. If x rises by 1 in one step and by 3 in another, a linear rule with slope 2 gives y changes of 2 and 6. Compare the ratios, not just the numbers in one row. For a table with x values 0, 2, 5 and y values 1, 5, 11, the ratios are 4/2 and 6/3; both are 2.
If the ratios disagree, inspect your copying and arithmetic first. The table may contain a typo, a deliberately non-linear example, or rounded measurements from the real world. Do not force a straight line through contradictory exact values merely because the exercise sits in a chapter on linear functions. State what the evidence supports and ask your teacher how to treat rounding if the source is measured data.
3. Draw a scale that fits every point
Before placing a dot, find the smallest and largest x and y values. Draw perpendicular axes and choose equal physical spacing for equal numerical steps along each axis. You can use different scales on x and y, but each individual axis must stay consistent. For our example, a scale of one square per unit comfortably includes x from −2 to 2 and y from −3 to 5.
Write a few tick labels clearly, including zero. Negative x values sit left of the vertical axis; negative y values sit below the horizontal one. A crowded graph can hide a sign error, so leave enough room around the edges and use a ruler for the axes. If your page is too small, choose two or five units per square and label that decision rather than squeezing unequal distances into the same space.
4. Plot two points, then use the third to check
Start with (0, 1): zero horizontal movement places it on the vertical axis, one unit above the origin. For (2, 5), move two units right and five units up. For (−2, −3), move two left and three down. Place small, precise dots. Saying the movements aloud can help prevent a sign from being lost between the table and graph.
Two distinct points determine one straight line, so draw through two and ask whether the third lies on it. The third is a useful independent check, not decorative extra work. In our example, the line through (0, 1) and (2, 5) also passes through (−2, −3). If it misses, inspect the ordered pairs, axis scale, and point locations before changing the equation.
A ruler helps, but do not bend the line toward a wrong point. For exact classroom values, all listed pairs should lie precisely on the same straight line when the table is linear. For measured data, points may scatter and a best-fit line is a different task. Read the instruction: 'draw the graph of the function' is not the same as 'estimate a trend from observations.'
5. Find slope from the table, not from a guess
Slope measures the change in y divided by the change in x between two distinct points. Using (0, 1) and (2, 5), the calculation is (5 − 1) / (2 − 0) = 4/2 = 2. The line rises two vertical units for each one horizontal unit. Choose another pair and check that it gives the same value. This is especially helpful when one row has a copying mistake.
Keep the subtraction order consistent. If you subtract y from the second point minus y from the first, subtract x in the same order. Reversing both signs leaves the ratio unchanged; reversing only one changes the answer's sign. For a descending line, the slope is negative: y decreases as x increases. A horizontal line has slope zero. Two rows with the same x but different y cannot describe one function at those inputs.
Slope has units in context. If x is hours and y is kilometres, a slope of 2 means two kilometres per hour, not simply '2'. Describing the units helps you catch an implausible interpretation. Serlo's explanation of linear functions connects a constant slope, the graph, and the formula. Use the same connection when you check your own table.
6. Identify the intercept and write an equation
In y = mx + b, m is the slope and b is the y value when x = 0. Our table includes the pair (0, 1), so b = 1. With m = 2, the equation is y = 2x + 1. Substitute x = −2: 2(−2) + 1 = −3. Substitute x = 2: 2(2) + 1 = 5. Both outputs agree with the table and therefore support the graph.
If the table does not show x = 0, use one known pair. For instance, if a line has slope 2 and includes (3, 7), put x = 3 and y = 7 into y = 2x + b. Then 7 = 6 + b, so b = 1. Do not label the first listed y value as the intercept just because it appears at the left of a table. The intercept belongs to x = 0, even when that input is omitted.
Different courses may write f(x) = mx + b or use another letter for the constant. The meaning is the same if the variables and units are defined. A graph can also show the intercept visually, but use substitution to confirm it. A plotted dot slightly off the axis can create a misleading reading, whereas the table and algebra can expose the error.
7. Capture the whole worksheet if you ask Lirno
When a photographed worksheet includes the table, graph grid, and written instructions, capture all of them rather than just the one cell you want explained. A cropped image can hide whether the task asks for a graph, an equation, or both. Keep the page flat, avoid glare, and make sure minus signs and axis labels are legible. If there are several questions, identify which one you are working on.
Before accepting a scan, compare the detected numbers with the page. In our example, confusing −2 with 2 or −3 with 3 changes the slope or intercept. Verify the row order as well. A table photographed sideways may be read as columns in the wrong direction. Correct or restate the values yourself if the scan is ambiguous; then request a hint, explanation, or check that matches your learning need.
Lirno is free to download and start, while some AI usage levels or advanced features may require Premium. That access detail does not change how you should study: use a hint before a complete explanation if you can still make an attempt. The scan is a way to provide context, not proof that an AI interpretation is correct. Follow your school's rules about permitted assistance and submitted work.
8. Ask for the first useful hint, then draw yourself
A good help request isolates the step you cannot perform: 'I have the pairs (−2, −3), (0, 1), and (2, 5). How can I check whether the rate of change is constant?' This invites a method instead of an unexplained final graph. If you already found slope 2, ask what x = 0 tells you. You can stop reading once you have enough information to continue independently.
Make your own graph before comparing it with an explanation. Choose the scale, place the points, draw the line, and write the equation on paper. If Lirno gives a different result, find the first disagreement: was a coordinate reversed, a minus sign lost, or an axis misread? A difference is more useful when you can name its cause than when you simply replace your answer with the app's output.
9. Check the graph against every row and its context
Use the equation as an independent test. For each x in the original table, calculate y = 2x + 1 and compare with the listed y. Then look at the plotted points. All three representations—table, equation, and graph—should tell the same story. A line that looks right but misses (2, 5) fails the check, even if the formula was written correctly nearby.
If this is a word problem, interpret b and m. A service with a fixed starting charge of one unit and a cost of two units per hour can fit y = 2x + 1, but only if negative hours are excluded. The algebraic line extends infinitely on paper; the meaningful domain may not. Label a restricted segment or explain the domain when the task concerns time, distance, money, or another quantity with practical limits.
A common checking mistake is to use a new point that you obtained from the same mistaken equation. That merely repeats the assumption. Check against a row from the original worksheet or a contextual fact supplied by the problem. If the row disagrees, revisit the transcription, slope computation, and intercept. Do not hide a mismatch under a thick line or an unexplained rounding claim.
10. Turn one graphing error into a short practice loop
After you fix a mistake, write the exact trigger in one sentence: 'I reversed x and y', 'I used uneven tick spacing', or 'I called the first y value the intercept.' Then make one new table with the same structure and different values. For example, try (−1, 4), (0, 2), and (1, 0). The y change is −2 for each unit of x, so the line slopes down and crosses the vertical axis at 2.
Cover the answer and repeat the process: ordered pairs, change ratios, axes, two plotted points, third-point check, slope, intercept, and substitution. A short independent retrieval attempt is more informative than rereading the explanation. If you used Lirno, ask it for a fresh practice prompt or check only after completing your version; verify that the new prompt is mathematically consistent before relying on it.
