Learning Center / Algebra

How to graph an equation: lines, parabolas and more

Three ways to graph an equation — a table of values, slope and intercept, and key points of a parabola — with worked examples and how to match a graph to its equation.

2 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

To graph an equation, find points (x,y)(x, y) that make the equation true, plot them on a coordinate grid, and join them with a line or a smooth curve. Every point on the graph is a solution of the equation, and every solution is a point on the graph — that is why graphing equations is such a useful way to see what an equation does.

This lesson covers the basic methods for equations in xx and yy: tables of values, lines and parabolas. When an equation is written as a function, f(x)f(x), the companion lesson on functions builds on these methods with domain, range and transformations.

Method 1: make a table of values

  1. Write the equation with yy on its own, if it isn’t already.
  2. Choose a few xx-values, such as −1,0,1,2-1, 0, 1, 2.
  3. Work out yy for each one.
  4. Plot the points and join them — a ruler for a straight line, a smooth curve otherwise.

Worked example

Easy: graph y=2x−1y = 2x - 1

Draw the graph of y=2x−1y = 2x - 1.

Show the working
  1. Make a table of values.
    x−1012y−3−113\begin{array}{c|cccc} x & -1 & 0 & 1 & 2 \\ y & -3 & -1 & 1 & 3 \end{array}
  2. Plot (−1,−3)(-1, -3), (0,−1)(0, -1), (1,1)(1, 1) and (2,3)(2, 3). They lie on a straight line.
  3. Check with two of the points: the slope is 3−(−1)2−0=2\frac{3 - (-1)}{2 - 0} = 2 and the line crosses the y-axis at −1-1.
    y=2x−1✓y = 2x - 1 \checkmark

Answer: A straight line through (0,−1)(0, -1) that rises 2 for every 1 to the right.

Checked by Sumzo’s math engine

Method 2: slope and y-intercept

An equation of the form y=mx+by = mx + b always gives a straight line — a linear relationship graph. Start at the y-intercept (0,b)(0, b), then use the slope mm as “rise over run” to find more points. For an equation like 3x+2y=63x + 2y = 6, it is often quicker to find where the line crosses each axis.

Worked example

Medium: sketch the graph of each line using intercepts

Sketch the graph of 3x+2y=63x + 2y = 6.

Show the working
  1. Put x=0x = 0: 2y=62y = 6, so the line crosses the y-axis at (0,3)(0, 3).
  2. Put y=0y = 0: 3x=63x = 6, so it crosses the x-axis at (2,0)(2, 0).
  3. Draw the line through both points. Its slope is 0−32−0=−32\frac{0 - 3}{2 - 0} = -\frac{3}{2}, so the line falls from left to right.
    y=−32x+3y = -\tfrac{3}{2}x + 3

Answer: The line through (0,3)(0, 3) and (2,0)(2, 0), which is y=−32x+3y = -\frac{3}{2}x + 3.

Checked by Sumzo’s math engine

Graphing a parabola

An equation with x2x^2, such as y=x2−4x+3y = x^2 - 4x + 3, makes a U-shaped curve called a parabola. Find three things before you draw: where it crosses the x-axis (set y=0y = 0), the y-intercept (set x=0x = 0), and the vertex, which is halfway between the x-intercepts.

Worked example

Hard: graph y=x2−4x+3y = x^2 - 4x + 3

Find the key points of y=x2−4x+3y = x^2 - 4x + 3 and sketch it.

Show the working
  1. x-intercepts: solve x2−4x+3=0x^2 - 4x + 3 = 0.
    (x−1)(x−3)=0⇒x=1 or x=3(x - 1)(x - 3) = 0 \Rightarrow x = 1 \text{ or } x = 3
  2. y-intercept: x=0x = 0 gives y=3y = 3.
    (0,3)(0, 3)
  3. Vertex: halfway between 1 and 3 is x=2x = 2, and y=4−8+3=−1y = 4 - 8 + 3 = -1.
    (2,−1)(2, -1)
  4. The x2x^2 term is positive, so the parabola opens upwards. Draw a smooth U through the points.

Answer: A parabola through (1,0)(1, 0), (3,0)(3, 0) and (0,3)(0, 3), with its lowest point at (2,−1)(2, -1).

Checked by Sumzo’s math engine

Which graph matches the equation?

  • Put x=0x = 0: the y-intercept must match.
  • For a line, check the sign of the slope: positive rises to the right, negative falls.
  • For a parabola, the sign of the x2x^2 term tells you whether it opens up or down.
  • Substitute one more point from the graph into the equation — it must make it true.

Common mistakes

  • Plotting (x,y)(x, y) the wrong way round: across first, then up.
  • Using only two points for a curve — a parabola needs at least five.
  • Joining the points of a parabola with straight lines instead of a smooth curve.
  • Uneven scales on the axes, which change how steep a line looks.

Frequently asked questions

How do you graph an equation step by step?

Get yy on its own, make a table of values, plot the points and join them. For a straight line two points are enough (use a third to check); for a curve use more.

How do I know if an equation is a straight line?

If it can be written as y=mx+by = mx + b — no x2x^2, no xx in a denominator, no square roots of xx — its graph is a straight line.

How do I graph an equation that isn’t solved for y?

Rearrange it first, or for a line like 3x+2y=63x + 2y = 6 find the two intercepts by setting x=0x = 0 and then y=0y = 0.

Calculators

Practice

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