HomeDrawing Linear and Quadratic Graphs: Plot, Compare, and Explain Free original lessons
Drawing Linear and Quadratic Graphs: Plot, Compare, and Explain
Students build value tables, plot a line and a parabola on shared axes, then compare intercepts, shape, and symmetry using checkable arithmetic.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
This lesson teaches you to draw a linear graph and a quadratic graph on the same coordinate axes from value tables, then compare their shape, intercepts, and symmetry. You will use simple whole-number inputs, consistent units, and a defined plotting convention so your work can be checked. The focus is on accurate plotting and clear comparison, not on solving quadratic equations by graphing or using vertex form. All numerical examples are hypothetical teaching values; measure your own grid and label it before plotting.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Building value tables for both functions
Choose a linear function such as y = 2x + 1 and a quadratic function such as y = x^2 - 2x - 3. Use the same input values for both tables so the comparison is fair. A practical input set is x = -2, -1, 0, 1, 2, 3, 4, but any consecutive integers that fit your grid are acceptable.
For the linear function, add the same amount to y for each step in x. For the quadratic, the second differences are constant when inputs are consecutive integers. Check each row by substitution before plotting. Label the table columns x, y, and working, and keep units consistent if the context uses them.
- Write the function rule at the top of the table.
- Substitute each chosen x value and record the y value.
- Check at least two rows by reversing the arithmetic or substituting back.
- Mark the ordered pairs clearly as (x, y) before plotting.
Plotting both graphs on shared axes
Draw one coordinate grid with equal spacing on both axes and label the scale. Plot the linear points first and join them with a straight line using a ruler. Plot the quadratic points and join them with a smooth curve; do not join quadratic points with straight segments.
Use a defined plotting convention: each point is a small cross centred on the intersection of the grid lines, and the curve passes through the centre of each cross. If your grid spacing differs from a hypothetical example, measure the actual spacing and state it. Valid alternatives include using dots instead of crosses, provided the centre is clear and consistent.
- Label the axes and state the scale in units per grid square.
- Plot each ordered pair using the agreed point convention.
- Join linear points with a ruled straight line.
- Draw the quadratic as a smooth curve through all plotted points.
- Label each graph with its function rule.
Comparing shape, intercepts, and symmetry
The linear graph has a constant slope and is a straight line. The quadratic graph is curved and opens upward when the coefficient of x^2 is positive. Compare where each graph crosses the y-axis by reading the y value when x = 0. Compare x-intercepts by reading where each graph crosses the x-axis, and check by substitution.
Symmetry applies to the quadratic: for y = x^2 - 2x - 3, the vertical line x = 1 is a line of symmetry because substituting x = 1 + h and x = 1 - h gives the same y value. The linear graph has no such symmetry. State comparisons as checkable sentences, for example: the line crosses the y-axis at (0, 1) and the parabola at (0, -3).
- Read and record the y-intercept of each graph.
- Read and record any x-intercepts visible on the grid.
- Test quadratic symmetry using two inputs equally spaced from the symmetry line.
- Write two comparison sentences with coordinates and arithmetic checks.
Worked example
Plot y = x + 2 and y = x^2 - 4 on shared axes for x = -2, -1, 0, 1, 2, then compare intercepts and symmetry.
Worked answer
Linear table: (-2, 0), (-1, 1), (0, 2), (1, 3), (2, 4). Quadratic table: (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0). The line crosses the y-axis at (0, 2); the parabola crosses at (0, -4). Both cross the x-axis at (-2, 0) and (2, 0) for the parabola, while the line crosses at (-2, 0). The parabola is symmetric about x = 0 because x = 1 and x = -1 both give y = -3; the line has no such symmetry.
Four practice exercises
Exercise 1
Plot y = 3x - 1 and y = x^2 - 1 on shared axes for x = -1, 0, 1, 2, 3. State both y-intercepts and one x-intercept of each graph.
Show answer and check
Linear table: (-1, -4), (0, -1), (1, 2), (2, 5), (3, 8). Quadratic table: (-1, 0), (0, -1), (1, 0), (2, 3), (3, 8). Linear y-intercept (0, -1); quadratic y-intercept (0, -1). Linear x-intercept at (1/3, 0) if the grid allows fractions, otherwise state it lies between 0 and 1; quadratic x-intercepts at (-1, 0) and (1, 0).
Why: Substitute each x into both rules. The shared y-intercept occurs because both rules give -1 at x = 0. The quadratic factors as (x - 1)(x + 1), giving x-intercepts at -1 and 1. The linear x-intercept is found by solving 3x - 1 = 0.
Check: Substitute the stated intercept coordinates back into the rules and confirm equality; re-read the plotted points against the tables.
Exercise 2
Plot y = -x + 4 and y = x^2 - 2x on shared axes for x = -1, 0, 1, 2, 3. Compare the shape and state whether the quadratic has a line of symmetry, with a substitution check.
Show answer and check
Linear table: (-1, 5), (0, 4), (1, 3), (2, 2), (3, 1). Quadratic table: (-1, 3), (0, 0), (1, -1), (2, 0), (3, 3). The line is straight with constant slope -1; the parabola opens upward. The quadratic is symmetric about x = 1 because x = 0 and x = 2 both give y = 0, and x = -1 and x = 3 both give y = 3.
Why: The linear rule decreases by 1 for each increase of 1 in x. The quadratic values repeat at equal distances from x = 1, confirming symmetry. The line has no repeated y values for different x values, so no vertical symmetry line.
Check: Substitute x = 1 + h and x = 1 - h for h = 1 and h = 2 into the quadratic and confirm equal y values; check the linear table differences are constant.
Exercise 3
Plot y = 0.5x + 1 and y = x^2 - 3 on shared axes for x = -2, -1, 0, 1, 2. State the y-intercepts and describe where each graph crosses the x-axis using the plotted points.
Show answer and check
Linear table: (-2, 0), (-1, 0.5), (0, 1), (1, 1.5), (2, 2). Quadratic table: (-2, 1), (-1, -2), (0, -3), (1, -2), (2, 1). Linear y-intercept (0, 1); quadratic y-intercept (0, -3). The line crosses the x-axis at (-2, 0). The parabola crosses the x-axis between -2 and -1 and between 1 and 2, since y changes sign in those intervals.
Why: Substitution gives the tables. The linear x-intercept is exact at x = -2 because 0.5(-2) + 1 = 0. The quadratic has no integer x-intercepts, so the plotted points show sign changes that locate the crossings between grid values.
Check: Confirm sign changes in the quadratic table between consecutive x values; substitute the linear intercept back into its rule.
Exercise 4
Plot y = 2x + 3 and y = x^2 + 2x - 3 on shared axes for x = -3, -2, -1, 0, 1. Compare the number of x-intercepts and check the quadratic symmetry line by substitution.
Show answer and check
Linear table: (-3, -3), (-2, -1), (-1, 1), (0, 3), (1, 5). Quadratic table: (-3, 0), (-2, -3), (-1, -4), (0, -3), (1, 0). The line has one x-intercept at (-1.5, 0) if fractions are used, otherwise between -2 and -1. The parabola has two x-intercepts at (-3, 0) and (1, 0). The quadratic is symmetric about x = -1 because x = 0 and x = -2 both give y = -3, and x = 1 and x = -3 both give y = 0.
Why: The quadratic factors as (x + 3)(x - 1), giving x-intercepts at -3 and 1. The linear rule gives one crossing because it is a straight line with nonzero slope. Equal y values at inputs equally spaced from x = -1 confirm the symmetry line.
Check: Substitute x = -1 + h and x = -1 - h for h = 1 and h = 2 into the quadratic; confirm equal y values. Substitute the linear intercept estimate back into y = 2x + 3.
Common mistakes
- Joining quadratic points with straight line segments. Draw a smooth curve through the plotted points and check that no sharp corners appear between points.
- Using different x values for the two tables, making comparison unreliable. Use the same input values in both tables unless a stated reason requires otherwise.
- Misreading the scale and plotting points one square off. State the scale in units per square and re-check two plotted points against the table.
- Claiming symmetry without testing values. Substitute two inputs equally spaced from the proposed symmetry line and confirm equal y values.
Self-check rubric
- Value tables are complete and checked by substitution.
- Axes are labelled and the scale is stated in units per square.
- Linear points are joined with a ruled straight line.
- Quadratic points are joined with a smooth curve through point centres.
- Intercepts are read correctly and stated as coordinates.
- Symmetry claim is supported by two equally spaced substitutions.
- Comparison sentences are clear and checkable.
- Valid alternative point markers are accepted if used consistently.
Questions and answers
Do I need to use vertex form for the quadratic?
No. This task uses value tables and plotting only. Vertex form is outside the boundary of this lesson.
Can I use decimals in the tables?
Yes, if the grid supports them, but keep units consistent and state the scale. Whole numbers reduce plotting errors.
How many points should I plot for the quadratic?
At least five points spanning both sides of the turning point, so the curve shape is visible and checkable.
What counts as a valid point marker?
A small cross or dot centred on the grid intersection, used consistently. The curve must pass through the centre of each marker.