TMUA: Roots, intersections and shape
Roots, intersections and the shape of graphs.
- Section 1 Part 1 · MM8 Graphs of functions · spec MM8.5-MM8.7
- Papers 1 and 2
- 8 practice questions
- No calculator
What the specification covers
- Stationary points and monotonicity from the derivative
- Possible numbers of real roots of a polynomial
- Counting intersections of two graphs = counting solutions
Key ideas
- The solutions of \(f(x)=g(x)\) are the \(x\)-coordinates where the graphs meet: count intersections on a sketch.
- A polynomial of degree \(n\) has at most \(n\) real roots; a cubic always has at least one.
- Use the stationary points to decide how many times a cubic crosses the axis: three roots need a maximum above and a minimum below the axis.
- Factorise in disguise: \(x^4-4x^3+4x^2=(x^2-2x)^2\).
Common mistakes
- A repeated root counts once as a 'distinct' root.
- \(|f(x)|\) reflects the parts below the axis upwards; it does not change the parts above.
Exam tip
For 'how many solutions' questions, a careful sketch is usually the whole answer.
Worked example
Worked example
For which value of \(k\) do the graphs of \(y=|x^2-4|\) and \(y=k\) meet at exactly three points?
- \(k > 4\)
- \(k = 2\)
- \(k = 0\)
- \(k = 4\)
- \(0 < k < 4\)
Answer: D: \(k = 4\)
\(y=|x^2-4|\) is \(x^2-4\) reflected in the axis between \(x=-2\) and \(x=2\), with a local maximum \((0,4)\). A horizontal line \(y=k\) meets it at 2 points (\(k=0\) or \(k>4\)), 4 points (\(0
Practice questions
Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.
Question 1
For which values of the constant \(k\) does the equation \(x^3-3x+k=0\) have three distinct real roots?
- \(0 < k < 2\)
- \(k < 2\)
- \(-2 \le k \le 2\)
- \(-2 < k < 2\)
- \(k > -2\)
Show the answer and solution
Answer: D: \(-2 < k < 2\)
Let \(f(x)=x^3-3x+k\). \(f'(x)=3x^2-3\), so the stationary points are at \(x=-1\) (local maximum, value \(k+2\)) and \(x=1\) (local minimum, value \(k-2\)). The cubic crosses the axis three times exactly when the maximum is above the axis and the minimum below it: \(k+2>0\) and \(k-2<0\), i.e. \(-2
Question 2
At how many points do the graphs of \(y=|x^2-1|\) and \(y=0\) meet?
- \(5\)
- \(4\)
- \(1\)
- \(3\)
- \(2\)
Show the answer and solution
Answer: E: \(2\)
\(y=|x^2-1|\) is \(x^2-1\) with the part between \(x=-1\) and \(x=1\) reflected upwards, giving a local maximum \((0,1)\). A horizontal line meets it twice if it is at height 0 or above \(1\), 3 times at height \(1\) and 4 times in between. Here the answer is 2.
Question 3
At how many points do the graphs of \(y=|x^2-4|\) and \(y=8\) meet?
- \(2\)
- \(1\)
- \(3\)
- \(4\)
- \(5\)
Show the answer and solution
Answer: A: \(2\)
\(y=|x^2-4|\) is \(x^2-4\) with the part between \(x=-2\) and \(x=2\) reflected upwards, giving a local maximum \((0,4)\). A horizontal line meets it twice if it is at height 0 or above \(4\), 3 times at height \(4\) and 4 times in between. Here the answer is 2.
Question 4
At how many points do the graphs of \(y=|x^2-9|\) and \(y=\frac{9}{2}\) meet?
- \(2\)
- \(1\)
- \(5\)
- \(4\)
- \(3\)
Show the answer and solution
Answer: D: \(4\)
\(y=|x^2-9|\) is \(x^2-9\) with the part between \(x=-3\) and \(x=3\) reflected upwards, giving a local maximum \((0,9)\). A horizontal line meets it twice if it is at height 0 or above \(9\), 3 times at height \(9\) and 4 times in between. Here the answer is 4.
Question 5
At how many points do the graphs of \(y=|x^2-4|\) and \(y=2\) meet?
- \(5\)
- \(1\)
- \(3\)
- \(4\)
- \(2\)
Show the answer and solution
Answer: D: \(4\)
\(y=|x^2-4|\) is \(x^2-4\) with the part between \(x=-2\) and \(x=2\) reflected upwards, giving a local maximum \((0,4)\). A horizontal line meets it twice if it is at height 0 or above \(4\), 3 times at height \(4\) and 4 times in between. Here the answer is 4.
Question 6
At how many points do the graphs of \(y=|x^2-9|\) and \(y=9\) meet?
- \(4\)
- \(5\)
- \(2\)
- \(3\)
- \(1\)
Show the answer and solution
Answer: D: \(3\)
\(y=|x^2-9|\) is \(x^2-9\) with the part between \(x=-3\) and \(x=3\) reflected upwards, giving a local maximum \((0,9)\). A horizontal line meets it twice if it is at height 0 or above \(9\), 3 times at height \(9\) and 4 times in between. Here the answer is 3.
Question 7
At how many points do the graphs of \(y=|x^2-1|\) and \(y=2\) meet?
- \(1\)
- \(4\)
- \(2\)
- \(5\)
- \(3\)
Show the answer and solution
Answer: C: \(2\)
\(y=|x^2-1|\) is \(x^2-1\) with the part between \(x=-1\) and \(x=1\) reflected upwards, giving a local maximum \((0,1)\). A horizontal line meets it twice if it is at height 0 or above \(1\), 3 times at height \(1\) and 4 times in between. Here the answer is 2.
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