C2.5, C3.2; E2.5, E3.2 - Solving equations graphically

Graphical solution of an equation

To solve $f(x)=g(x)$, graph $y=f(x)$ and $y=g(x)$. The solutions are the $x$-coordinates of their intersections. The $y$-coordinates confirm equality but are not extra solutions for $x$.

Worked example 1: a quadratic and a line

Question. Use a graphical method to solve $x^2=3x+4$.

Plan. Find intersections of the graphs of the two sides.

1. Enter $Y_1=X^2$ and $Y_2=3X+4$.
2. Use $-3\le x\le6$, $-6\le y\le30$. Sketch the upward curve and rising line, showing their two meetings.
3. Use intersection near $-1$ and 4. The points are $(-1,1)$ and $(4,16)$.

Answer. $x=-1$ or $x=4$.

Check. At $-1$, both sides equal 1; at 4, both equal 16. The two intersection inputs give exactly the required equality.

Worked example 2: an unfamiliar curve

Question. Use a graphical method to solve $x^3=x+2$ for $0\le x\le3$, correct to 3 significant figures.

Plan. Graph both sides over the stated input interval.

1. Enter $Y_1=X^3$, $Y_2=X+2$, with $0\le x\le3$, $0\le y\le28$.
2. The curve meets the line once between 1 and 2. Zoom there while keeping the original interval in mind.
3. The intersection tool gives approximately $(1.521379707,3.521379707)$.

Answer. $x=1.52$, to 3 significant figures.

Check. A table for the difference gives $1.521^3-1.521-2\approx-0.00226$ and $1.522^3-1.522-2\approx0.00369$. The crossing is between them. The full-domain graph shows no other meeting in $0\le x\le3$.

Watch out

For “solve the equation”, give the intersection inputs. Do not report the two coordinates of one intersection as two different solutions.

Difference function

$f(x)=g(x)$ is equivalent to $f(x)-g(x)=0$, wherever both original expressions are defined. Graph the difference and locate its zeros.

Worked example 3: moving everything to one side

Question. Use a GDC zero feature to solve $x^2+x=7$, correct to 3 significant figures.

Plan. Find the zeros of $h(x)=x^2+x-7$.

1. Enter $Y_1=X^2+X-7$, using $-5\le x\le4$, $-9\le y\le16$.
2. Isolate the crossings in $(-4,-3)$ and $(2,3)$.
3. The zero feature gives approximately $-3.192582404$ and $2.192582404$.

Answer. $x=-3.19$ or $x=2.19$, to 3 significant figures.

Check. Substituting the full calculator values in $x^2+x$ gives 7 in each case. With rounded inputs both give 6.9861, close to 7 as expected.

Worked example 4: a touching zero

Question. Solve $(x-1)^2=0$ graphically.

Plan. Examine the lowest point as well as looking for crossings.

1. Enter $Y_1=(X-1)^2$, using $-2\le x\le4$, $-1\le y\le10$.
2. The curve touches the axis near $x=1$. If a zero search needs a sign-changing crossing and fails here, use the permitted minimum feature near 1 instead.
3. The minimum is $(1,0)$, so its input is a zero.

Answer. $x=1$, the single distinct solution.

Check. $(1-1)^2=0$. At $x=0.9$ and 1.1 the outputs are both 0.01, so a sign-change search alone would miss this touch.