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Date May 2013 Marks available 2 Reference code 13M.1.sl.TZ1.2
Level SL only Paper 1 Time zone TZ1
Command term Write down Question number 2 Adapted from N/A

Question

The diagram below shows part of the graph of \(f(x) = (x - 1)(x + 3)\) .


(a)     Write down the \(x\)-intercepts of the graph of \(f\) .

(b)     Find the coordinates of the vertex of the graph of \(f\) .

[6]
.

Write down the \(x\)-intercepts of the graph of \(f\) .

[2]
a.

Find the coordinates of the vertex of the graph of \(f\) .

[4]
b.

Markscheme

(a)     \(x = 1\) , \(x = - 3\) (accept (\(1\), \(0\)), (\( - 3\), \(0\)) )     A1A1     N2

[2 marks]

 

(b)     METHOD 1

attempt to find \(x\)-coordinate     (M1)

eg   \(\frac{{1 + - 3}}{2}\) , \(x = \frac{{ - b}}{{2a}}\) , \(f'(x) = 0\)

correct value, \(x = - 1\) (may be seen as a coordinate in the answer)     A1

attempt to find their \(y\)-coordinate     (M1)

eg   \(f( - 1)\) , \( - 2 \times 2\) , \(y = \frac{{ - D}}{{4a}}\)

\(y = - 4\)     A1

vertex (\( - 1\), \( - 4\))     N3  

METHOD 2

attempt to complete the square     (M1)

eg   \({x^2} + 2x + 1 - 1 - 3\) 

attempt to put into vertex form     (M1)

eg   \({(x + 1)^2} - 4\) , \({(x - 1)^2} + 4\)

vertex (\( - 1\), \( - 4\))     A1A1     N3

[4 marks]

.

\(x = 1\) , \(x = - 3\) (accept (\(1\), \(0\)), (\( - 3\), \(0\)) )     A1A1     N2

[2 marks]

 

a.

METHOD 1

attempt to find \(x\)-coordinate     (M1)

eg   \(\frac{{1 + - 3}}{2}\) , \(x = \frac{{ - b}}{{2a}}\) , \(f'(x) = 0\)

correct value, \(x = - 1\) (may be seen as a coordinate in the answer)     A1

attempt to find their \(y\)-coordinate     (M1)

eg   \(f( - 1)\) , \( - 2 \times 2\) , \(y = \frac{{ - D}}{{4a}}\)

\(y = - 4\)     A1

vertex (\( - 1\), \( - 4\))     N3  

METHOD 2

attempt to complete the square     (M1)

eg   \({x^2} + 2x + 1 - 1 - 3\) 

attempt to put into vertex form     (M1)

eg   \({(x + 1)^2} - 4\) , \({(x - 1)^2} + 4\)

vertex (\( - 1\), \( - 4\))     A1A1     N3

[4 marks]

b.

Examiners report

Most candidates recognized the values of the x-intercepts from the factorized form of the function. Candidates also showed little difficulty finding the vertex of the graph, and employed a variety of techniques: averaging \(x\)-intercepts, using \(x = \frac{{ - b}}{{2a}}\) , completing the square.

.

Most candidates recognized the values of the x-intercepts from the factorized form of the function. Candidates also showed little difficulty finding the vertex of the graph, and employed a variety of techniques: averaging \(x\)-intercepts, using \(x = \frac{{ - b}}{{2a}}\) , completing the square.

a.

Most candidates recognized the values of the x-intercepts from the factorized form of the function. Candidates also showed little difficulty finding the vertex of the graph, and employed a variety of techniques: averaging \(x\)-intercepts, using \(x = \frac{{ - b}}{{2a}}\) , completing the square.

b.

Syllabus sections

Topic 2 - Functions and equations » 2.4 » The quadratic function \(x \mapsto a{x^2} + bx + c\) : its graph, \(y\)-intercept \((0, c)\). Axis of symmetry.
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