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

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IB Math SL Chapter 1: Lines

• Equations of straight lines
• Parallel and perpendicular lines

Warm-Up

1. Points A(1,2) and B(4,-7).
   Equation in the form y=mx+b?
   Solution
   y=-3x+5
   
   
2. 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
   
   
3. 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

1. Sketch y=x^2-4
   Solution
   
   
   
2. Sketch y=(x-2)^2
   Solution
   
   
   
3. 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
      
      
4. 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
      
      
5. 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)
      
      
6. What is the equation of the quadratic below in the form:
   y=ax^2+bx+c?
   
   Solution
   y=2(x-2)^2-2
   
   
7. 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}
      
      
8. Solve h(t)=-3t^2-9t
   Solution
   0, -3
   
   
9. Solve points of intersections of the simultaneous equations:
   y=2x^2-x and y-2=2x
   Solution
   (-0.5,1) and (2,6)
   
   
10. 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
    
    
11. 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}
    
    
12. 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)
    
    
13. f(x)=x^2+2x-3. How far away is the point (2,0) from the vertex?
    Solution
    5 units
    
    
14. 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
       
       
15. 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}
       
       
16. g(x)=x^2+bx+11. P(-1,8) is on the graph of g. Find b.
    Solution
    4
    
    
17. Find k for which the equations 2x^2+6x+k=0 has repeated roots.
    Solution
    \frac{9}{2}
    
    
18. 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
    
    
19. 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
    
    
20. f(x)=2(x+2)(x-4). y=kx-16 is tangent to this parabola. Find k.
    Solution
    -4
    
    
21. 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
       
       
22. 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
23. 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
       
       
24. \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
25. 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}
26. f(x)=(x-2)^2, x\geq2. Find f^{-1}(x)
27. f(x)=x^2+2x-3, x<-1.
    1. Find f^{-1}{x}
    2. Evaluate f(-3)
    3. Evaluate f^{-1}(0)
28. 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)

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