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1  $(\mathrm{IB} / \mathrm{sl} / 2016 / \mathrm{May} /$ paper $1 \mathrm{tz} 1 / \mathrm{q} 7)$
[Maximum mark: 8]
Let $f(x)=3 \tan ^{4} x+2 k$ and $g(x)=\tan ^{4} x+8 k \tan ^{2} x+k$, for $0 \leq x \leq 1$, where $0 < k < 1 .$
The graphs of $f$ and $g$ intersect at exactly one point. Find the value of $k$.

2  (IB/s1/2019/November/Paper1/q3)
[Maximum mark: 7 ]
Let $g(x)=x^{2}+b x+11$. The point $(1,8)$ lies on the graph of $g$.
(a)  Find the value of $b$. 
(b)  The graph of $f(x)=x^{2}$ is transformed to obtain the graph of $g$. Describe this transformation. [4] 

3  $(\mathrm{IB} / \mathrm{s} / / 2019 / \mathrm{May} /$ paper $1 \mathrm{tz} 1 / \mathrm{q} 8)$
[Maximum mark: 16]
Let $f(x)=9x^{2}, x \in \mathbb{R}$.
(a)  Find the $x$intercepts of the graph of $f$. 
The following diagram shows part of the graph of $f$.
Rectangle $\mathrm{PQRS}$ is drawn with $\mathrm{P}$ and $\mathrm{Q}$ on the $x$axis and $\mathrm{R}$ and $\mathrm{S}$ on the graph of $f$. Let $\mathrm{OP}=b$.
(b)  Show that the area of $\mathrm{PQRS}$ is $18 b2 b^{3}$. 
(c)  Hence find the value of $b$ such that the area of PQRS is a maximum. 
Consider another function $g(x)=(x3)^{2}+k, x \in \mathbb{R}$.
(d)  Show that when the graphs of $f$ and $g$ intersect, $2 x^{2}6 x+k=0$. 
(e)  Given that the graphs of $f$ and $g$ intersect only once, find the value of $k$. [5] 

4  (IB/s1/2019/May/paper 1tz2/q5)
[Maximum mark: 6]
Consider the function $f(x)=(1k) x^{2}+x+k, x \in \mathbb{R}$. Find the value of $k$ for which $f(x)$ has two equal real roots.

5  (IB/s1/2019/May/paper2tz1/q2)
[Maximum mark: 6]
Consider the graph of the function $f(x)=\alpha(x+10)^{2}+15, x \in \mathbb{R}$.
(a)  Write down the coordinates of the vertex. 
(b)  The graph of $f$ has a $y$intercept at $20$. Find $a$. [2] 
(c)  Point $\mathrm{P}(8, b)$ lies on the graph of $f$. Find $b$. [2] 

6  (IB/sl/2018/November/Paper1/q8)
[Maximum mark: 16]
Let $f(x)=x^{2}4 x5$. The following diagram shows part of the graph of $f$.
(a)  Find the $x$intercepts of the graph of $f$. 
(b)  Find the equation of the axis of symmetry of the graph of $f$. 
(c)  The function can be written in the form $f(x)=(xh)^{2}+k$.
(i) Write down the value of $h$.
(ii) Find the value of $k$. $[4]$ 
The graph of a second function, $g$, is obtained by a reflection of the graph of $f$ in the $y$axis, followed by a translation of $\left(\begin{array}{c}3 \\ 6\end{array}\right)$.
(d)  Find the coordinates of the vertex of the graph of $g$. [5] 

7  (IB/sl/2018/May/paper1tz1/q4)
[Maximum mark: 7]
Let $f(x)=a x^{2}4 xc$. A horizontal line, $L$, intersects the graph of $f$ at $x=1$ and $x=3$.
(a)  (i) The equation of the axis of symmetry is $x=p$. Find $p$.
(ii) Hence, show that $a=2$. 
(b)  The equation of $L$ is $y=5$. Find the value of $c$. 

8  (IB/sl/2018/May/paper1tz2/q6)
[Maximum mark: 7 ]
Let $f(x)=p x^{2}+q x4 p$, where $p \neq 0$. Find the number of roots for the equation $f(x)=0$. Justify your answer.

9  (IB/sl/2018/May/paper2tz1/q4)
[Maximum mark: 7]
Let $g(x)=(x1)^{2}+5$.
(a)  Write down the coordinates of the vertex of the graph of $g$. $[1]$ 
Let $f(x)=x^{2}$. The following diagram shows part of the graph of $f$.
The graph of $g$ intersects the graph of $f$ at $x=1$ and $x=2$.
(b)  On the grid above, sketch the graph of $g$ for $2 \leq x \leq 4$. 
(c)  Find the area of the region enclosed by the graphs of $f$ and $g$. [3] 

10  (IB/s1/2017/November/Paper1/q8)
[Maximum mark: 16]
Let $f(x)=x^{2}x$, for $x \in \mathbb{R}$. The following diagram shows part of the graph of $f$.
The graph of $f$ crosses the $x$axis at the origin and at the point $\mathrm{P}(1,0)$.
(a)  Show that $f^{\prime}(1)=1$. 
The line $L$ is the normal to the graph of $f$ at $\mathrm{P}$.
(b)  Find the equation of $L$ in the form $y=a x+b$. 
The line $L$ intersects the graph of $f$ at another point $Q$, as shown in the following diagram.
(c)  Find the $x$ coordinate of $Q$. 
(d)  Find the area of the region enclosed by the graph of $f$ and the line $L$. 

11  (IB/s1/2017/May/paper1tz1/q9)
[Maximum mark: 14]
A quadratic function $f$ can be written in the form $f(x)=a(xp)(x3)$. The graph of $f$ has axis of symmetry $x=2.5$ and $y$intercept at $(0,6)$.
(a)  Find the value of $p$. 
(b)  Find the value of $a$. 
(c)  The line $y=k x5$ is a tangent to the curve of $f$. Find the values of $k$. [8] 

12  (IB/sl/2017/May/paper2tz2/q6)
[Maximum mark: 8]
Let $f(x)=x^{2}1$ and $g(x)=x^{2}2$, for $x \in \mathbb{R}$.
(a)  Show that $(f \circ g)(x)=x^{4}4 x^{2}+3$. 
(b)  On the following grid, sketch the graph of $(f \circ g)(x)$, for $0 \leq x \leq 2.25$. 
(c)  The equation $(f \circ g)(x)=k$ has exactly two solutions, for $0 \leq x \leq 2.25$. Find the possible values of $k$. 

13  (IB/sl/2016/November/Paper1/q1)
[Maximum mark: 6]
Let $f(x)=x^{2}4 x+5$
(a)  Find the equation of the axis of symmetry of the graph of $f$. 
The function can also be expressed in the form $f(x)=(xh)^{2}+k$.
(b)  (i) Write down the value of $h$.
(ii) Find the value of $k$. 

14  (IB/s1/2016/November/Paper2/q1)
[Maximum mark: 7 ]
Let $f(x)=x^{2}+2 x+1$ and $g(x)=x5$, for $x \in \mathbb{R}$.
(b)  Find $(g \circ f)(x)$. [2] 
(c)  Solve $(g \circ f)(x)=0$. 

15  (IB/s1/2016/May/paper1tz1/q5)
[Maximum mark: 6]
Consider $f(x)=x^{2}+q x+r$. The graph of $f$ has a minimum value when $x=1.5$. The distance between the two zeros of $f$ is $9 .$
(a)  Show that the two zeros are 3 and $6$. 
(b)  Find the value of $q$ and of $r$, 

16  (IB/s1/2016/May/paper1tz2/q1)
[Maximum mark: 6]
The following diagram shows part of the graph of a quadratic function $f$. The vertex is at $(3,1)$ and the $x$intercepts at 2 and 4 . The function $f$ can be written in the form $f(x)=(xh)^{2}+k$.
(a)  Write down the value of $h$ and of $k$. 
The function can also be written in the form $f(x)=(xa)(xb)$
(b)  Write down the value of $a$ and of $b$. [2] 
(c)  Find the $y$intercept. [2] 

17  (IB/sl/2015/November/Paper1/q8)
[Maximum mark: 16]
The following diagram shows part of the graph of a quadratic function $f$.
The vertex is at $(1,9)$, and the graph crosses the $y$axis at the point $(0, c)$.
The function can be written in the form $f(x)=(xh)^{2}+k$.
(a)  Write down the value of $h$ and of $k$, [2] 
(b)  Find the value of $c$. [2] 
Let $g(x)=(x3)^{2}+1$. The graph of $g$ is obtained by a reflection of the graph of $f$ in the $x$axis, followed by a translation of $\left(\begin{array}{l}p \\ q\end{array}\right)$.
(c)  Find the value of $p$ and of $q$  
(d)  Find the $x$coordinates of the points of intersection of the graphs of $f$ and $g$. $[7]$ 

18  (IB/sl/2015/May/paper1tz1/q6)
[Maximum mark: 6]
Let $f(x)=p x^{2}+(10p) x+\frac{5}{4} p5$.
(a)  Show that the discriminant of $f(x)$ is $1004 p^{2}$ 
(b)  Find the values of $p$ so that $f(x)=0$ has two equal roots. 

19  (IB/sl/2015/May/paper1tz2/q8)
[Maximum mark: 15]
Let $f(x)=a(x+3)(x1)$. The following diagram shows part of the graph of $f .$
The graph has $x$intercepts at $(p, 0)$ and $(q, 0)$, and a $y$intercept at $(0,12)$
(a)  (i) Write down the value of $p$ and of $q$.
(ii) Find the value of $a$, [6] 
(b)  Find the equation of the axis of symmetry of the graph of $f$. [3] 
(c)  Find the largest value of $f$. [3] 
The function $f$ can also be written as $f(x)=a(xh)^{2}+k$
(d)  Find the value of $h$ and of $k$. [3] 

Answer
$1 k=\frac{1}{4}$
2 (a) 4 (b) translation, $\left(\begin{array}{c}2 \\ 7\end{array}\right)$
3 (a) $x=\pm 3$ (b) Show
(b) Show (c) $\sqrt{3}$
(d) Show (e) $4.5$
4 $k=\frac{1}{2}$
5 (a) $(10,15)$
(b) $0.35$ (c) $98.4$
6 (a) $x=1, x=5(b) \quad x=2$ (c) (i) $h=2$ (ii) $k=9$
(d) $(5,3)$
7 (a)(i) $p=1$ (ii) Show (b) $c=1$
8 f has 2 roots
9 (a) $(1,5)$ (b) Graph (c) $A=9$
10 (a) Show
(b) $y=x+1$
(c) $x=1$
(d) Area $=\frac{4}{3}$
11 (a) $\quad p=2$ (b) $a=1$
(c) $k=3,7$
12 (a) Show (b) Graph (c) $1<k \leqslant 3$
13 (a) $x=2$ (b)(i) $\quad h=2$ (ii) $k=1$
14 (a) 81 (b) $(g \circ f)(x)=x^{2}+2 x4$
(c) $x=1.24, x=3.24$
15 (a) Show (b) $q=3, r=18$
16 (a) $h=3, \quad k=1$ (b) $a=2, b=4$ (or $a=4, b=2)$ (c) $y=8$
17(a) $\quad h=1, k=9$ (b) $c=8$ (c) $p=2, q=8$ (d) $x=0, x=4$
18(a) Show (b) $p=\pm 5$
19(a) (i) $p=3, q=1$ (ii) $a=4$ (b) $x=1$ (c) 16 (d) $h=1, k=16$
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