Page 59 - Math 9-10 (2009)

58
THE ONTARIO CURRICULUM, GRADES 9 AND 10: MATHEMATICS
Quadratic Relations of the Form y
=
ax2 + bx + c
Overall Expectations
By the end of this course, students will:
• manipulatealgebraicexpressions,asneededtounderstandquadraticrelations; • identifycharacteristicsofquadraticrelations;
• solveproblemsbyinterpretinggraphsofquadraticrelations.
Specific Expectations
Manipulating Quadratic Expressions
By the end of this course, students will:
– expand and simplify second-degree polyno- mial expressions involving one variable that consist of the product of two binomials [e.g., (2x + 3)(x + 4)] or the square of a binomial [e.g., (x + 3)2], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g. patterning);
– factor binomials (e.g., 4x2 + 8x) and trino- mials (e.g., 3x2 + 9x – 15) involving one variable up to degree two, by determining a common factor using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g., patterning);
– factor simple trinomials of the form
x2 + bx + c (e.g., x2 + 7x + 10,
x2 + 2x – 8), using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g., patterning);
– factor the difference of squares of the form x2 – a2 (e.g., x2 – 16).
Identifying Characteristics of Quadratic Relations
By the end of this course, students will:
– collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology (e.g., concrete materials, scientific probes, graphing calculators), or from secondary sources (e.g., the Internet, Statistics Canada); graph the data and draw a curve of best fit,
if appropriate, with or without the use of technology (Sample problem: Make a 1 m ramp that makes a 15° angle with the floor. Place a can 30 cm up the ramp. Record the time it takes for the can to roll to the bot- tom. Repeat by placing the can 40 cm,
50 cm, and 60 cm up the ramp, and so on. Graph the data and draw the curve of best fit.);
– determine, through investigation using technology, that a quadratic relation of the form y = ax2 + bx + c (a ≠ 0) can be graph- ically represented as a parabola, and deter- mine that the table of values yields a con- stant second difference (Sample problem: Graph the quadratic relation y = x2 – 4, using technology. Observe the shape of the graph. Consider the corresponding table of values, and calculate the first and second differences. Repeat for a different quadratic relation. Describe your observations and make conclusions.);
– identify the key features of a graph of a parabola (i.e., the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum value), using a given graph or a graph generated with technology from its equation, and use the appropriate terminol- ogy to describe the features;
– compare, through investigation using tech- nology, the graphical representations of a quadratic relation in the form
y=x2 +bx+candthesamerelationin the factored form y = (x – r)(x – s) (i.e., the graphs are the same), and describe the