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Intersections of a line and a quadratic graph

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Graph showing the parabola y equals (x plus 1) squared minus 2 and the straight line y equals x plus 2 crossing at two points-4-224-4-22460xy
Curved edge
\(\displaystyle y=(x+1)^2-2\)
Straight guide
\(\displaystyle y=x+2\)

A road-design plan shows the curved edge and straight guide given below. A crossing point lies on both graphs. Which equation locates the x-values of those crossing points?

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

Intersecting a line and a parabola

An intersection is a point that satisfies both rules at the same time. Equate the two outputs, solve the resulting quadratic in x, then use a rule to find each matching y-value

\[\Delta=b^2-4ac\]
  1. Equate the outputs At a common point, the quadratic output and the line output are equal \(\displaystyle Q(x)=L(x)\)
  2. Use zero form Move all terms to one side before identifying the quadratic coefficients \(\displaystyle x^2-3x+1-(2x+1)=0\Rightarrow x^2-5x=0\)
  3. Read the discriminant A negative, zero or positive discriminant gives zero, one or two real common x-values \(\displaystyle \Delta=(-5)^2-4(1)(0)=25>0\)
  4. Pair each x with y Substitute each x-value into one rule and check the same y-value in the other rule \(\displaystyle x=5\Rightarrow y=2(5)+1=11\)

Worked example

Find the common points of the line and parabola shown

\[y=-x+4,\qquad y=x^2-5x+4\]
  1. \(\displaystyle x^2-5x+4=-x+4\)At a common point, set the two y-expressions equal
  2. \(\displaystyle x^2-4x=0\)Move all terms to the left and simplify
  3. \(\displaystyle x(x-4)=0\Rightarrow x=0\text{ or }x=4\)Factorise and apply the null factor law
  4. \(\displaystyle L(0)=4,\quad L(4)=0\Rightarrow (0,4)\text{ and }(4,0)\)Use the line rule to find the matching y-coordinate for each x-value

Common mistakes

Solving the graphs separately
✕\(\displaystyle x^2-3x+1=0\)✓\(\displaystyle x^2-3x+1=2x+1\)
An intersection must satisfy both equations at the same x-value, so solving either graph on its own does not enforce a shared point
Using the uncombined discriminant
✕\(\displaystyle \Delta=(-3)^2-4(1)(1)=5\)✓\(\displaystyle \Delta=(-5)^2-4(1)(0)=25\)
The discriminant must be calculated from the quadratic formed after the two graph rules are equated
Reversing discriminant classification
✕\(\displaystyle \Delta=-4\Rightarrow 2\text{ common points}\)✓\(\displaystyle \Delta=-4<0\Rightarrow 0\text{ common points}\)
A negative discriminant means the combined quadratic has no real x-value and therefore no common point

Difficulty levels

introEquate a line and a parabola with two integer intersection points, then find both coordinates
standardUse the discriminant to recognise a tangent point or no common point
challengeHandle rational coefficients or exact irrational intersection x-values
examCombine the rules, classify the intersections and give coordinates when the values are rational

Questions about this topic

Why are the two graph outputs equated?+

At a common point both graphs have the same x-value and the same y-value. Equating their y-rules removes the shared output and leaves an equation in x

What does the discriminant tell me about the graphs?+

For the combined quadratic, a negative discriminant gives no real intersections, zero gives one tangent intersection, and a positive discriminant gives two intersections

How do I find the y-coordinate after solving for x?+

Substitute each x-value into the line rule or the quadratic rule. The resulting y-value is the coordinate paired with that x-value

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 3: Quadratic functions

In this chapter

Factorise, solve and reconstruct a rational quadraticCalculate and interpret a quadratic discriminantTransform a quadratic and find its key featuresComplete the square and link equivalent formsDetermine a quadratic rule from exact informationSolve a quadratic exactly with the quadratic formulaConstruct a rational quadratic from a surd rootSolve an equation that reduces to quadratic formAnalyse signs and solve a quadratic inequalityIntersections of a line and a quadratic graphDetermine parameter values from a discriminant conditionFormulate and solve a quadratic problemAnalyse and optimise a quadratic model