IB Math SL Analysis and Approaches

Here you will find original math practice problems aligned with the IB Math SL analysis and approaches curriculum. Visit this site directly at hunkim.com/sl

IB Math SL Overview

Lines
Quadratic Functions
Exponents and Logarithms
Rational Functions
Binomial Theorem
Trigonometry
Statistics
Calculus

IB Math SL Chapter 1: Lines

Equations of straight lines
Parallel and perpendicular lines

Warm-Up

Points $A(1,2)$ and $B(4,-7)$ .
Equation in the form $y=mx+b$ ?
Solution
$y=-3x+5$
$L_1: 3y-2x=4$
$L_2$ is perpendicular to $L_1$ and has an x-intercept of 4. Find $L_2$ in the form $y=mx+b$ .
Solution
$y=-\frac{3}{2}x+6$
$f(x)=e^x$ . The slope at $x=2$ is $e^2$ .
1. Find the equation of the tangent line at $x=2$ in the form $f(x)=mx+b$ .
  Solution
  $f(x)=e^2 x-e^2$
2. $y$ -intercept of the tangent line?
  Solution
  $-e^2$

IB Math SL Chapter 2: Quadratics

Solving quadratic equations and inequalities
Discriminant
Domain, range, inverse
Composite functions, identity
Graphing and transformation functions

Warm-Up

Sketch $y=x^2-4$
Solution
Sketch $y=(x-2)^2$
Solution
$f(x)=(x+1)^2-4$
1. Sketch
  Solution
2. Coordinates of the vertex?
  Solution
  $(-1,-4)$
3. Domain?
  Solution
  $x\in\mathbb{R}$
4. Range?
  Solution
  $y\geq -4$
5. Evaluate $f(2)$
  Solution
  $5$
6. Given $x>2$ find the range of $f(x)$
  Solution
  $y>5$
$y=-2(x-3)^2+8$
1. Sketch
  Solution
2. Write in the form $y=ax^2+bx+c$
  Solution
  $y=-2x^2+12x-10$
3. Value of the discriminant?
  Solution
  $64$
4. Find the domain in which $f(x)>6$
  Solution
  $2<x<4$
$f(x)=2(x-2)(x+4)$
1. Intercepts?
  Solution
  $2, -4$
2. Equation of the line of symmetry?
  Solution
  $x=-1$
3. Coordinates of the vertex?
  Solution
  $V(-1,-18)$
What is the equation of the quadratic below in the form:
$y=ax^2+bx+c$ ?

Solution
$y=2(x-2)^2-2$
Solve $y=2x^2-5x+3$
1. By factoring
  Solution
  $1, \frac{3}{2}$
2. By using the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
  Solution
  $1, \frac{3}{2}$
3. By completing the square
  Solution
  $1, \frac{3}{2}$
Solve $h(t)=-3t^2-9t$
Solution
$0, -3$
Solve points of intersections of the simultaneous equations:
$y=2x^2-x$ and $y-2=2x$
Solution
$(-0.5,1)$ and $(2,6)$
$f(x)=x^2-4x+3$ . Describe the transformation to move this parabola’s vertex to the origin.
Solution
Translate 2 units left and 1 unit up
Write in vertex form: $y=\frac{2}{3}x^2-\frac{x}{2}+1$
Solution
$y=\frac{2}{3}\left(x-\frac{3}{8}\right)^2+\frac{29}{32}$
Given $f(x)=(x-2)^2+4$ what are the coordinates of the maximum of $\frac{1}{f(x)}$ ?
Solution
$\left(2,\frac{1}{4}\right)$
$f(x)=x^2+2x-3$ . How far away is the point $(2,0)$ from the vertex?
Solution
5 units
$f(x)=2px^2+(p-8)x+\frac{3}{5}p-1$
1. Show that the discriminant is $-\frac{19}{5}p^2-8p+64$
2. Find the values of $p$ so that $f(x)=0$ has two equal roots
  Solution
  $p\approx -5.29, 3.18$
Solve:
1. $x^2\leq 9$
  Solution
  $-3\leq x\leq 3$
2. $x^2>4$
  Solution
  $x<-2$ or $x>2$
3. $x^2<5$
  Solution
  $-\sqrt{5}<x<\sqrt{5}$
4. $(x-2)^2<25$
  Solution
  $-3<x<7$
5. $(2x+1)^2\geq 7$
  Solution
  $x\leq \frac{\sqrt{7}-1}{2}$ or $x\geq \frac{\sqrt{2}-1}{2}$
$g(x)=x^2+bx+11$ . $P(-1,8)$ is on the graph of $g$ . Find $b$ .
Solution
$4$
Find $k$ for which the equations $2x^2+6x+k=0$ has repeated roots.
Solution
$\frac{9}{2}$
$f(x)=(k-2)x^2+x+k$ . Find the value(s) of $k$ for which $f(x)$ has no real roots.
Solution
$-0.12<k<2.12$
$f(x)=hx^2+kx-4h, h>0$ . Find the number of roots for the equation $f(x)=0$ . Justify your answer.
Solution
2 solutions because $D>0$
$f(x)=2(x+2)(x-4)$ . $y=kx-16$ is tangent to this parabola. Find $k$ .
Solution
$-4$
$f(x)=ax^2-4x-c$ . A horizontal line, $L$ , intersects the graph of $f$ at $x=-1$ and $x=3$ .
1. Find the axis of symmetry.
  Solution
  $x=1$
2. Find $a$ .
  Solution
  $a=2$
3. The equation of $L$ is $y=5$ . Find the value of $c$ .
  Solution
  $c=-1$
$f(x)=ax^2-12x+c$
1. A horizontal line, $L$ , intersects $f$ at $x=2$ and $x=4$
  1. Find the equation of the line of symmetry
    Solution
    $x=3$
  2. Hence show that $a=2$
2. The equation of $L$ is $y=-2$ . Find $c$ .
  Solution
  $14$
See $y=ax^2+bx+c$ below:

Positive, negative, or zero?
1. $a$
  Solution
  Negative
2. $b$
  Solution
  Negative
3. $c$
  Solution
  $0$
4. $b^2-4ac$
  Solution
  $0$
$\text{Revenue}=\text{num items}\times \text{price per item}.$ You normally sell 200 items at $50 each. For each $1 price increase you lose 2 sales.
1. Define $x$ given $R=(200-2x)(50+x)$
  Solution
  # of price increases
2. Maximum revenue?
  Solution
  $\$11250$
3. What price should you sell each item to maximize revenue?
  Solution
  $\$75$
The perimeter of the diagram below is 40.
1. Show that $y=20-\frac{\pi x}{2}$
2. Show that the area $A=20x-\frac{5\pi x^2}{16}$
$f(x)=(x-2)^2, x\geq2$ . Find $f^{-1}(x)$
$f(x)=x^2+2x-3, x<-1$ .
1. Find $f^{-1}{x}$
2. Evaluate $f(-3)$
3. Evaluate $f^{-1}(0)$
$f(x)=x-3$ and $g(x)=x^2$ .
1. Find $f(g(x))$
2. Find $(g\circ f)(x)$
3. Find $(f\circ g)^{-1}(x)$