Equation from a graph
Read the intercept, calculate the gradient and place both in the equation
\[y=mx+c\]
- Intercept Read the vertical coordinate at the vertical-axis crossing \(\displaystyle (0,c)\)
- Gradient Use signed coordinate changes in the same direction \(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)
- Notation The order of the terms may change without changing their roles \(\displaystyle mx+c=c+mx\)
- Check Substitute another visible point into the completed equation
Worked example
Use the vertical-axis crossing and the other supplied point to find the equation
\[P(0,-5),\quad Q(6,4)\]
- \(\displaystyle c=-5\)The first point is on the vertical axis
- \(\displaystyle m=\frac{4-(-5)}{6-0}\)Use the vertical change divided by the horizontal change
- \(\displaystyle m=\frac{9}{6}\)Evaluate the two changes
- \(\displaystyle m=\frac{3}{2}\)Write the gradient as an exact fraction
- \(\displaystyle y=\frac{3}{2}x-5\)Place the two coefficients in the equation
- \(\displaystyle \begin{aligned}\frac{3}{2}(6)-5&=9-5\\&=4\end{aligned}\)The second point's vertical coordinate is reproduced
Common mistakes
Swapping the coefficients
✕\(\displaystyle y=-5x+\frac{3}{2}\)✓\(\displaystyle y=\frac{3}{2}x-5\)
The gradient multiplies the variable; the intercept is the constantInverting the changes
✕\(\displaystyle m=\frac{6}{9}\)✓\(\displaystyle m=\frac{9}{6}\)
The vertical change belongs in the numeratorDifficulty levels
introRecognise a rule from a stated gradient and intercept
standardComplete the gradient and intercept in a supplied form from a graph, including horizontal lines
challengeComplete the coefficients using exact third and quarter gradients and a visible intercept
examRecognise a complete equation from a graph without intermediate prompts
Questions about this topic
Is the left edge the vertical axis?+
Only when the horizontal coordinate there is zero. Use the scale, not the frame
What if the intercept is outside the picture?+
Use two points to find the gradient, then substitute a point to recover the intercept. The later two-point topic combines those steps
Can the gradient be zero?+
Yes. A horizontal line has the same vertical coordinate at every point
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 5: Linear functions and models
In this chapter
Substitute values into a formulaConstruct a table of values from a formulaRecognise linear equationsSolve a linear equationSet up and solve a practical linear equationBuild and use a formula in two unknownsRead coordinates and points from a straight-line graphClassify and compare gradients from graphs5D Straight-line graphs and slope: Recognise a line or find points for its graph5D Straight-line graphs and slope: Find the gradient of a line through two points5E Straight-line equations: Recognise a line or find points for its graph5E Straight-line equations: Find the gradient of a line through two pointsRead the vertical-axis intercept and gradient from an equation5E Straight-line equations: Find the equation of a line from its graph or from its intercept and slope5F Equations from graphs and points: Find the equation of a line from its graph or from its intercept and slopeFind the equation of a line through two pointsConstruct a linear model from a descriptionInterpret the graph of a linear modelRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSet up a pair of simultaneous equations from a descriptionSolve a practical problem with simultaneous equationsUse a piecewise linear modelRead and construct step graphs