Sec 3 · G1 (N(T))

Graphs of functions

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1. Solving equations graphically

Where the curve meets the line

\displaystyle x^{2} - 2x - 3 = x + 1

Curve y = x² − 2x − 3 and line y = x + 1 meeting where x = −1 and x = 4.

e.g.

\displaystyle x^{2} - 2x - 3 = x + 1

\displaystyle x = -1 \text{ or } x = 4

Which line to draw: make the drawn curve one side

\displaystyle x^{2} - 3x - 4 = 0

e.g.

\displaystyle x^{2} - 3x - 4 = 0

\displaystyle x^{2} - 2x - 3 = x + 1

the curve $y = x^{2} - 2x - 3$ is drawn, so draw $y = x + 1$

📖 Worked example

The diagram shows the graph of $y = x^{2} - 2x$ and the line $y = 3$ . Use the graph to solve the equation $x^{2} - 2x = 3$ .

Step 1 of 3

Each side of the equation is a graph

Left-hand side:

y = x^{2} - 2x

, the curveRight-hand side:

y = 3

, the horizontal line

📖 Worked example

The graph of $y = -x^{2} + 2x + 3$ has been drawn. Find the equation of the straight line that must be drawn on the same axes to solve the equation $-x^{2} + 5x + 6 = 0$ .

Step 1 of 3

Rearrange so the drawn curve is the left-hand side

-x^{2} + 5x + 6 = 0

-x^{2} + 2x + 3 = -3x - 3

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1. Solving equations graphically

Where the curve meets the line

\displaystyle x^{2} - 2x - 3 = x + 1

e.g.

\displaystyle x^{2} - 2x - 3 = x + 1

\displaystyle x = -1 \text{ or } x = 4

Which line to draw: make the drawn curve one side

\displaystyle x^{2} - 3x - 4 = 0

e.g.

\displaystyle x^{2} - 3x - 4 = 0

\displaystyle x^{2} - 2x - 3 = x + 1

the curve $y = x^{2} - 2x - 3$ is drawn, so draw $y = x + 1$

📖 Worked example

The diagram shows the graph of $y = x^{2} - 2x$ and the line $y = 3$ . Use the graph to solve the equation $x^{2} - 2x = 3$ .

Step 1 of 3

Each side of the equation is a graph

Left-hand side:

y = x^{2} - 2x

, the curveRight-hand side:

y = 3

, the horizontal line

📖 Worked example

The graph of $y = -x^{2} + 2x + 3$ has been drawn. Find the equation of the straight line that must be drawn on the same axes to solve the equation $-x^{2} + 5x + 6 = 0$ .

Step 1 of 3

Rearrange so the drawn curve is the left-hand side

-x^{2} + 5x + 6 = 0

-x^{2} + 2x + 3 = -3x - 3

Practise this →