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Self-Assessment for Grade 12 College Technology Math (MCT4C)
Self-Assessment for Grade 12 College Technology Math (MCT4C)

Students who are registered for Grade 12 College Technology Math (MCT4C) may benefit from a self evaluation and review of the following sample of expectations from Grade 11 University/College Math (MCF3M).


 The questions in this self-assessment reflect some of the key ideas learned in prerequisite courses.  They do not represent the problem solving approach or the rich experience that students would be exposed to in a classroom.  The intention is for students to revisit some key concepts and, if needed, access review materials in an informal environment at a pace that is comfortable for the student. 



Concept

Sample Question and Answer

How comfortable do you feel with this concept? 

Link for further support

I can solve quadratic equations

1. Solve by factoring

a) x squared minus x minus 2 equals 0

b) negative 14 x squared plus 7 x equals 0

c) x squared minus 81 equals 0

Very comfortable, somewhat comfortable, or not at all comfortable 

Solving Quadratic Equations

I can determine the real roots of a variety of quadratic equations (i.e., graphing; factoring; using the quadratic formula) 

2. Determine the roots of each quadratic.


a) x squared plus 6 x plus 8 equals 0

b) 3 x plus 3 equals 5 x squared

Very comfortable, somewhat comfortable, or not at all comfortable 

Introduction to the Quadratic Formula

I can sketch graphs of g of x equals a left parenthesis x – h right parenthesis squared plus k by applying one or more transformations to the graph of f of x equals x squared

3. For the quadratic relation  y equals negative open parentheses x plus 3 close parentheses squared plus 4

a) Graph the Relation


State the:

b) direction of the opening;                    

c) coordinates of the vertex;                

d) equation of the axis of symmetry; 

e) y-intercept;

f) x-intercepts.

Very comfortable, somewhat comfortable, or not at all comfortable 

Transformations of Parabolas 

I can sketch graphs of quadratic functions in the factored form f of x equals a left parenthesis x – r right parenthesis left parenthesis x – s right parenthesis by using the x-intercepts to determine the vertex

4. Sketch the function 

f of  x equals 2 open parentheses x minus 1 close parentheses open parentheses x minus 5 close parentheses 

Very comfortable, somewhat comfortable, or not at all comfortable 

Graphing and Equations in Factored Form

I can solve problems arising from real-world applications, given the algebraic representation of a quadratic function.

5. Given the equation of a quadratic function representing the height of a ball over elapsed time h of t  equals negative 0.75 t squared plus 15 t

a) Determine the maximum height of the ball.

b) Determine the length of time needed for the ball to touch the ground.

c) Determine the time interval when the ball is higher than 40m.

Very comfortable, somewhat comfortable, or not at all comfortable 

Quadratic Applications

I can evaluate, with and without technology, numerical expressions containing integer and rational exponents and rational bases

6. Evaluate

a) open parentheses 2 to the power of negative 1 end exponent close parentheses squared

b)open parentheses negative 27 close parentheses to the power of 1 third end exponent


open parentheses 25 over 4 close parentheses to the power of 3 over 2 end exponent

c) 



Very comfortable, somewhat comfortable, or not at all comfortable 

Exponent Laws all Together

I can describe key properties for exponential functions ( domain and range, intercepts, increasing/decreasing intervals, and asymptotes) 

7. Describe key properties for 


f of x equals 200 open parentheses 1 half close parentheses to the power of x 

Very comfortable, somewhat comfortable, or not at all comfortable 

Properties of Basic Exponential Functions

I can graph, with and without technology, an exponential relation, given its equation in the form y equals a to the power of x comma open curly brackets a greater than 0 comma a not equal to 1 close curly brackets

8. Graph the function 

f of x equals 200 open parentheses 1 half close parentheses to the power of x 

Very comfortable, somewhat comfortable, or not at all comfortable 

Transformations of Exponential Functions

I can solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications by interpreting the graphs 

9. New Zealand has a population of about 6 million and it is estimated that the population will double in 36 years, modelled by P of t equals P subscript 0 open parentheses 2 close parentheses to the power of t over 36 end exponent


If population growth remains the same, what will the population be in 15 years? 40 years?

Very comfortable, somewhat comfortable, or not at all comfortable 

Modelling with Exponential Functions

I can solve real-world application problems by using the primary trigonometric ratios, by determining the measures of the sides and angles of right triangles

10. To measure the height of a tall tree, a forester paces 40 m from the base of the tree and measures an angle of elevation to the top of the tree.  If the angle is 65 degrees, determine the height of the cedar tree.

Very comfortable, somewhat comfortable, or not at all comfortable 

Sine and Cosine Ratios

I can solve problems that require the use of the sine law or the cosine law in acute triangles.

11. Two ships left Port Huron on Lake Ontario at the same time.  One travelled at 12 km/h on a course of 235 degrees.  The other travelled at 15km/h on a course of 105 degrees. How far apart are they four  hours later?

Very comfortable, somewhat comfortable, or not at all comfortable 

The Sine Law

I can sketch graphs of f of x equals a sin left parenthesis x – d right parenthesis plus cby applying the transformation to the graph of f of x equals sin x and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/ decreasing intervals)

12. Consider f of  x equals 0.5 sin open parentheses x minus 30 close parentheses plus 1


a) Sketch the graph over the interval  0 to 360 degrees.


b) Describe its key properties.

Very comfortable, somewhat comfortable, or not at all comfortable 

Investigate Transformations of Sinusoidal Functions


I can identify periodic and sinusoidal functions

13. Which of the following scenarios could be modelled by periodic functions?


a) Sunset times in Aurora, Ontario.


b) The motion of a planet around the sun.


c) A child’s movement on a swing.


d) The growth of a bacteria. 

Very comfortable, somewhat comfortable, or not at all comfortable 

Modelling Periodic Behaviour

Solutions to Sample Questions


1. Solve by factoring

a) x squared minus x minus 2 equals 0

b) negative 14 x squared plus 7 x equals 0

c) x squared minus 81 equals 0


Solutions: 


a) open parentheses x minus 2 close parentheses open  parentheses x plus 1 close parentheses equals 0 space rightwards arrow space x minus 2 space equals space 0 space or space x space plus space 1 space equals space 0 space rightwards arrow space x space equals space 2 comma space minus 1


7 x open parentheses negative 2 x plus 1 close parentheses space equals space 0 space rightwards arrow space x space equals space 0 space or space minus 2 x plus 1 space equals space 0 space rightwards arrow space x space equals space 0 comma fraction numerator space 1 over denominator 2 end fraction

b)

 

c) open parentheses x minus 9 close parentheses open parenthesis x plus 9 right parentheses space equals space 0 space rightwards arrow space x space equals space 9 comma space minus 9




2. Determine the roots of each quadratic.

a)x squared plus 6 x plus 8 equals 0

b) 3 x plus 3 equals 5 x squared


Solutions:

a) open parentheses x space plus space 4 close parentheses open parentheses space x space plus space 2 close parentheses space equals space 0 space rightwards arrow space x space equals space minus 4 comma space minus 2

b) 

 4 Lines. Line 1: x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction. Line 2:  x equals fraction numerator negative open parentheses negative 3 close parentheses plus-or-minus square root of open parentheses negative 3 close parentheses squared minus 4 open parentheses 5 close parentheses open parentheses negative 3 close parentheses end root over denominator 2 open parentheses 5 close parentheses end fraction. line 3: x equals fraction numerator 3 plus-or-minus square root of 9 plus 60 end root over denominator 10 end fraction. line 4: x equals space minus 0.53 comma space 1.13




3. For the quadratic relation  y equals negative open parentheses x plus 3 close parentheses squared plus 4

a) Graph the Relation


State the:

b) direction of the opening;                    

c) coordinates of the vertex;                

d) equation of the axis of symmetry; 

e) y-intercept;

f) x-intercepts.


Solutions:


  1. Graph of the required parabola as seen in desmos 


b)  up

c) open parentheses negative 3 comma space 4 close parentheses

d) x equals negative 3

e) open parentheses 0 comma space 13 close parentheses

f) there are no x-intercepts for this function.



4. Sketch the function 

f of x equals 2 open parentheses x minus 1 close parentheses open parentheses x minus 5 close parentheses 


Solution: 


Use the x-intercepts to plot.  Sketch the axis of symmetry as x equals 3. Locate the vertex using substitution of x equals 3 rightwards arrow f of 3 equals negative 8.  Connect intercepts and vertex to sketch.


Sketch of required parabola as seen on desmos 




5. Given the equation of a quadratic function representing the height of a ball over elapsed time h of t  equals negative 0.75 t squared plus 15 t

a) Determine the maximum height of the ball.

b) Determine the length of time needed for the ball to touch the ground.

c) Determine the time interval when the ball is higher than 40m.


Solutions:

Sketch of given parabola as seen on desmos 

a) Max height 75m

b) 20 seconds

c)

Using interpolation, an additional feature of vertical lines connected to where the y value is 40 is shown. 

Using  interpolation - it appears that the height is greater than 40m from approximately 3 seconds to approximately 17 seconds, for a total of approximately 14 seconds.


Alternatively, one could also solve the equation  negative 0.75 t squared plus 15 t equals 40 to get 13.7 seconds.




6. Evaluate

a) open parentheses 2 to the exponent negative 1 close parentheses squared

b)open parentheses negative 27 close parentheses to the exponent 1 third


c) open parentheses 25 over 4 close parentheses to the exponent 3 over 2 end exponent



Solutions: 


2 to the exponent negative 2 end exponent equals open parentheses 2 over 1 close parentheses to the exponent negative 2 end exponent equals open parentheses 1 half close parentheses squared which equals 1 fourth

a)

 

b)cube root of negative 27 end root equals negative 3

 


c)   open parentheses 25 over 4 close parentheses to exponent 3 over 2 end exponent equals open parentheses square root of 25 over 4 end root close parentheses cubed equals open parentheses 5 over 2 close parentheses cubed equals 125 over 8


or


open parentheses 25 over 4 close parentheses to the exponent 3 over 2 end exponent equals square root of open parentheses 25 over 4 close parentheses cubed end root equals open parentheses 5 over 2 close parentheses cubed equals 125 over 8 




7. Describe key properties for 


f of x equals 200 open parentheses 1 half close parentheses to the exponent x 


Solution:

decay curve.

decreasing function .

Rate of decay = 1 half.

y-intercept at 200.



f of x equals 200 open parentheses 1 half close parentheses to the exponent x


8. Graph the function

Solution: 

Graph of required function as seen on desmos 




9. New Zealand has a population of about 6 million and it is estimated that the population will double in 36 years, modelled by P of t equals P naught open parentheses 2 close parentheses to the exponent of t over 36 .  If population growth remains the same, what will the population be in 15 years? 40 years?


Solutions:

a) P of t equals 6 open parentheses 2 close parentheses to the exponent 15 over 36 end exponent equals 8.009million

b) P of t equals 6 open parentheses 2 close parentheses to the exponent 40 over 36 end exponent equals 12.9607 million



10. To measure the height of a tall tree, a forester paces 40 m from the base of the tree and measures an angle of elevation to the top of the tree.  If the angle is 65 degrees, determine the height of the cedar tree.


Solution:

Right triangle containing given information the angle of elevation from horizontal side is given as 65 degrees.  The length of horizontal side is 40m. The vertical side length is labelled by the unknown variable, h. 

3 Lines. Line 1: tan 65 degrees equals h over 40. Line 2:   h equals 40 tan 65 degrees. Line 3:  h equals 86 


Therefore the height of the tree is 86m.




11. Two ships left Port Huron on Lake Ontario at the same time.  One travelled at 12 km/h on a course of 235 degrees.  The other travelled at 15km/h on a course of 105 degrees. How far apart are they four hours later?


Solution:

Diagram of given information on a cartesian plane Two directions are labelled using the bearings of 105 degrees and 235 degrees. 


From the diagram, the obtuse angle formed between the directions of the two ships is 130 degrees.

After four hours of travel, the ships have travelled respectively 48 km and 60km.


Obtuse triangle with angle 130 degrees and two adjacent side lengths of 48 and 60. 

3 Lines. Line 1: a squared equals 48 squared plus 60 squared - 2 open parentheses 48 close parentheses open parentheses 60 close parentheses cosine 130 degrees. line 2:  equals 9606.45.  line 3:  a = 98 


Therefore the two ships are 98km apart after 4 hours.




12. Consider

f of  x equals 0.5 sin open parentheses x minus 30 close parentheses plus 1 


a) Sketch the graph over the interval  0 degrees less than or equal to x less than or equal to 360 degrees.


b) Describe its key properties.


Solutions:


a) Sketch of sine wave as seen on desmos



b) Amplitude = 0.5.  Period length = 360 degrees. Phase shift of 30 degreesright. Vertical translation of 1 unit up. Maximum at 1.5.  Minimum at 0.5.



13. Which of the following scenarios could be modelled by periodic functions?


a) Sunset times in Aurora, Ontario


b) The motion of a planet around the sun


c) A child’s movement on a swing


d) The growth of a bacteria 



Solutions:


a and b only.




 
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