What do the intersections of y=f(x) and y=g(x) represent?

Solutions of f(x)=g(x) (the x-coordinates).

After finding x at an intersection, how do you find y?

Substitute x into either equation.

Number of solutions of f(x)=k means what graphically?

How many times y=k meets y=f(x).

Another name for finding where a line meets a curve?

Solving simultaneous equations.

Using graphs to solve equationsAQA A-Level Maths: Flashcards

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What is the graphical meaning of a solution of $f(x)=0$?

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What is the graphical meaning of a solution of $f(x)=0$ ?: An $x$ -intercept of $y=f(x)$ .
What do the intersections of $y=f(x)$ and $y=g(x)$ represent?: Solutions of $f(x)=g(x)$ (the $x$ -coordinates).
After finding $x$ at an intersection, how do you find $y$ ?: Substitute $x$ into either equation.
A curve touches the $x$ -axis. What does this show?: A repeated root.
To solve $x^2-4x-3=0$ using $y=x^2-4x-5$ , which line?: $y=-2$ .
What line solves $x^2-3x-10=0$ using $y=x^2-2x-8$ ?: $y=x+2$ .
A line is a tangent to a curve. What is the discriminant of the combined quadratic?: $b^2-4ac=0$ .
A line does not meet a curve. What is the discriminant of the combined quadratic?: $b^2-4ac<0$ .
A line cuts a curve in two points. What is the discriminant of the combined quadratic?: $b^2-4ac>0$ .
Number of solutions of $f(x)=k$ means what graphically?: How many times $y=k$ meets $y=f(x)$ .
For $y=x^3-3x$ , how many solutions has $x^3-3x=k$ when $|k|<2$ ?: Three.
For $y=x^3-3x$ , for which $k$ is there exactly one solution?: $k>2$ or $k<-2$ .
Another name for finding where a line meets a curve?: Solving simultaneous equations.

Exam questions on Using graphs to solve equations

A student draws the curve $y=x^2-4x-5$ and the straight line $y=2x-5$ on the same axes.
The student now wants to solve $x^2-4x-3=0$ using the same curve. Find the equation of the straight line that should be drawn.2 marks
The graph of $y=x^3-3x$ is drawn, with a horizontal line $y=k$ on the same axes. The curve has a maximum at $(-1,2)$ and a minimum at $(1,-2)$ .
State the values of $k$ for which the equation $x^3-3x=k$ has exactly one real solution.2 marks
A curve has equation $y=x^2-5x+2$ and a line has equation $y=x-7$ .
Show that the $x$ -coordinates of any points where the line meets the curve satisfy $x^2-6x+9=0$ , and explain what this tells you about the line.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).