Basic factored form sketching

Questions asking to sketch a polynomial curve given explicitly in factored form like y = (x-a)(x-b)(x-c) or y = (x-a)²(x-b), showing intercepts and shape, without additional transformations or follow-up parts.

9 questions · Moderate -0.8

1.02n Sketch curves: simple equations including polynomials
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OCR C1 2005 January Q5
7 marks Easy -1.2
5 On separate diagrams,
  1. sketch the curve \(y = \frac { 1 } { x }\),
  2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
  3. sketch the curve \(y = - \sqrt { } x\).
OCR C1 2008 June Q6
6 marks Moderate -0.8
6
  1. Expand and simplify \(( x - 5 ) ( x + 2 ) ( x + 5 )\).
  2. Sketch the curve \(y = ( x - 5 ) ( x + 2 ) ( x + 5 )\), giving the coordinates of the points where the curve crosses the axes.
OCR MEI C1 Q1
3 marks Easy -1.8
1 Sketch the graph of \(y = 9 - x ^ { 2 }\).
OCR MEI FP2 2011 June Q5
18 marks Standard +0.8
5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).
  1. On separate diagrams, sketch the curve in each of the following cases.
    (A) \(m = 1 , n = 1\),
    (B) \(m = 2 , n = 2\),
    (C) \(m = 2 , n = 4\),
    (D) \(m = 4 , n = 2\).
  2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
  3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
  4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
  5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
  6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. }{www.ocr.org.uk}) after the live examination series.
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OCR MEI C1 2007 January Q2
3 marks Easy -1.8
2 Sketch the graph of \(y = 9 - x ^ { 2 }\).
Edexcel C3 Q34
9 marks Standard +0.3
The function \(f\) is even and has domain \(\mathbb{R}\). For \(x \geq 0\), \(f(x) = x^2 - 4ax\), where \(a\) is a positive constant.
  1. In the space below, sketch the curve with equation \(y = f(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
  2. Find, in terms of \(a\), the value of \(f(2a)\) and the value of \(f(-2a)\). [2]
Given that \(a = 3\),
  1. use algebra to find the values of \(x\) for which \(f(x) = 45\). [4]
OCR PURE Q6
7 marks Easy -1.2
Sketch the following curves.
  1. \(y = \frac{2}{x}\) [2]
  2. \(y = x^3 - 6x^2 + 9x\) [5]
SPS SPS SM 2025 October Q6
3 marks Moderate -0.8
Sketch the graph of $$y = (x - k)^2(x + 2k)$$ where \(k\) is a positive constant. Label the coordinates of the points where the graph meets the axes. \includegraphics{figure_6} [3]
OCR H240/02 2018 December Q2
5 marks Moderate -0.3
\(\text{f}(x)\) is a cubic polynomial in which the coefficient of \(x^3\) is 1. The equation \(\text{f}(x) = 0\) has exactly two roots.
  1. Sketch a possible graph of \(y = \text{f}(x)\). [2]
It is now given that the two roots are \(x = 2\) and \(x = 3\).
  1. Find, in expanded form, the two possible polynomials \(\text{f}(x)\). [3]