Quadratic Function (IB SL)

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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 $f(x)=9-x^{2}, x \in \mathbb{R}$.
(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 $18 b-2 b^{3}$.
(c) Hence find the value of $b$ such that the area of PQRS is a maximum.

Consider another function $g(x)=(x-3)^{2}+k, x \in \mathbb{R}$.
(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 $f(x)=(1-k) x^{2}+x+k, x \in \mathbb{R}$. 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 $f(x)=x^{2}-4 x-5$. 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 $f(x)=(x-h)^{2}+k$.
(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 $f(x)=a x^{2}-4 x-c$. 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 $f(x)=p x^{2}+q x-4 p$, 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 $g(x)=-(x-1)^{2}+5$.
(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 $f(x)=a(x-p)(x-3)$. 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 $y=k x-5$ 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 $f(x)=(x-h)^{2}+k$.
(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 $g(x)=x-5$, for $x \in \mathbb{R}$.
(a) Find $f(8)$. [2]
(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 $f(x)=(x-h)^{2}+k$.
(a) Write down the value of $h$ and of $k$.

The function can also be written in the form $f(x)=(x-a)(x-b)$
(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 $f(x)=(x-h)^{2}+k$.
(a) Write down the value of $h$ and of $k$, [2]
(b) Find the value of $c$. [2]

Let $g(x)=-(x-3)^{2}+1$. 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 $f(x)=p x^{2}+(10-p) x+\frac{5}{4} p-5$.
(a) Show that the discriminant of $f(x)$ is $100-4 p^{2}$
(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 $f(x)=a(x+3)(x-1)$. 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 $f(x)=a(x-h)^{2}+k$
(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) $(g \circ f)(x)=x^{2}+2 x-4$
(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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