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Approximate solutions from intersections of two graphs

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Solve with graph intersections

The solutions of P(x)=Q(x) are the horizontal coordinates of the intersections of the two graphs. Use either rule with an unrounded x-value to obtain the matching vertical coordinate. Use the equation to verify exact values or justify finer rounding

\[P(x)=Q(x)\Longleftrightarrow P(x)-Q(x)=0\]
  1. 1Set the outputs equal At an intersection, both rules have the same output for the same input \(\displaystyle P(x)=Q(x)\Longleftrightarrow P(x)-Q(x)=0\)
  2. 2Use horizontal coordinates as solutions Each intersection in the stated window gives one solution through its x-coordinate \(\displaystyle P(a)=Q(a)\Rightarrow x=a\text{ is a solution}\)
  3. 3Recover the vertical coordinate Substitute the unrounded intersection x-coordinate into either original rule. Retain guard digits and round each requested coordinate only at the end \(\displaystyle x=a\Rightarrow y=P(a)=Q(a)\)
  4. 4Match the graph precision Use the graph to locate intersections. Verify exact values and resolve uncertain rounding with the displayed rules or a suitable numerical method \(\displaystyle \text{one-unit grid}\not\Rightarrow\text{two decimal places}\)
Year 11, VCE Units 1 & 2, Chapter 5, 4 difficulty levels, Calculator allowedBuild a worksheet

Graphical polynomial solutions

The solutions of P(x)=Q(x) are the horizontal coordinates of the intersections of the two graphs. Use either rule with an unrounded x-value to obtain the matching vertical coordinate. Use the equation to verify exact values or justify finer rounding

\[P(x)=Q(x)\Longleftrightarrow P(x)-Q(x)=0\]
  1. Set the outputs equal At an intersection, both rules have the same output for the same input \(\displaystyle P(x)=Q(x)\Longleftrightarrow P(x)-Q(x)=0\)
  2. Use horizontal coordinates as solutions Each intersection in the stated window gives one solution through its x-coordinate \(\displaystyle P(a)=Q(a)\Rightarrow x=a\text{ is a solution}\)
  3. Recover the vertical coordinate Substitute the unrounded intersection x-coordinate into either original rule. Retain guard digits and round each requested coordinate only at the end \(\displaystyle x=a\Rightarrow y=P(a)=Q(a)\)
  4. Match the graph precision Use the graph to locate intersections. Verify exact values and resolve uncertain rounding with the displayed rules or a suitable numerical method \(\displaystyle \text{one-unit grid}\not\Rightarrow\text{two decimal places}\)

Worked example

Find the intersection points of the two polynomial rules below

\[P(x)=x^2-3x+5,\qquad Q(x)=x+2\]
  1. \(\displaystyle P(x)=Q(x)\Rightarrow x^2-3x+5=x+2\Rightarrow x^2-4x+3=0\)Equate the two outputs and move all terms to one side
  2. \(\displaystyle x^2-4x+3=\left(x-1\right)\left(x-3\right)=0\Rightarrow x=1\text{ or }x=3\)Factor the difference and use the null factor law to find the horizontal coordinates
  3. \(\displaystyle Q(1)=1+2=3\Rightarrow\left(1,3\right),\qquad Q(3)=3+2=5\Rightarrow\left(3,5\right)\)Substitute each solution into Q to obtain the corresponding vertical coordinate
  4. \(\displaystyle P(1)=3=Q(1),\qquad P(3)=5=Q(3)\)The equal values verify both intersection points in the original rules

Common mistakes

Solving the rules separately
✕\(\displaystyle P(x)=0\Rightarrow x=-1\quad\text{and}\quad Q(x)=0\Rightarrow x=-\frac{1}{3}\)✓\(\displaystyle P(x)=x^2+2x+1,\qquad Q(x)=3x+1\Rightarrow P(x)=Q(x)\Rightarrow x^2+2x+1=3x+1\)
An intersection is found by equating the two outputs, not by solving each rule's zeros separately
Reporting only horizontal coordinates
✕\(\displaystyle x=1\text{ and }x=3\)✓\(\displaystyle \left(1,3\right)\text{ and }\left(3,5\right)\)
An intersection point includes both its horizontal and vertical coordinates
Reporting unsupported precision
✕\(\displaystyle \text{one-unit grid}\Rightarrow x=1.00\)✓\(\displaystyle \text{one-unit grid}\not\Rightarrow\text{certified decimal places}\)
Grid spacing alone does not certify exact coordinates or rounding near a boundary. Use the equation to check the requested precision

Difficulty levels

introMatching graph pairs and rational intersections are identified and counted
standardIntersection coordinates are reported to the requested precision, using the rules to check uncertain graph readings
challengeA rearrangement is chosen and numerical refinement distinguishes sign-changing crossings from contacts
examA symbolic choice, intersection table, solution count and precision judgement are combined

Questions about this topic

Why does an intersection solve P(x)=Q(x)?+

At an intersection both graphs have the same x-coordinate and the same y-coordinate, so their outputs are equal for that input

How do you find the y-coordinate after reading an intersection?+

Substitute the unrounded x-coordinate, or retain sufficient guard digits, in either original rule. Round y only at the end; using an already rounded x can change its rounded value

Why can a graph give only an approximate solution?+

The grid and viewing window limit how precisely a coordinate can be read. Use the equation to justify finer precision. A sign-changing bracket can refine a crossing, but an even-multiplicity contact needs a repeated-root solver or separate exact verification

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, Mathematical Methods, chapter 5: Higher-degree polynomials

In this chapter

Interpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityAnalyse a parameterised polynomial familyBracket and approximate a polynomial root by bisectionApproximate solutions from intersections of two graphsLocate a polynomial turning point numerically