Page 48 - Math 9-10 (2009)

PRINCIPLES OF MATHEMATICS, GRADE 10, ACADEMIC (MPM2D) 47
Quadratic Relations of the Form y
=
ax2 + bx + c
Overall Expectations
By the end of this course, students will:
• determine the basic properties of quadratic relations;
• relate transformations of the graph of y = x2 to the algebraic representation y=a(x–h)2 +k;
• solve quadratic equations and interpret the solutions with respect to the corresponding relations;
• solve problems involving quadratic relations.
Specific Expectations
Investigating the Basic Properties 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 bottom. 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 with
and without the use of technology, that a quadratic relation of the form
y=ax2 +bx+c(a≠0)canbegraphically represented as a parabola, and that the table of values yields a constant second difference (Sample problem: Graph the relation
y = x2 – 4x by developing a table of
values and plotting points. Observe the shape of the graph. Calculate first and second differences. Repeat for different quadratic relations. Describe your observa- tions and make conclusions, using the appropriate terminology.);
– 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 maxi- mum or minimum value), and use the appropriate terminology to describe them;
– compare, through investigation using tech- nology, the features of the graph of y = x2 and the graph of y = 2x, and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for y = 2x; by applying the exponent rules for multipli- cation and division).
=
By the end of this course, students will:
– identify, through investigation using tech- nology, the effect on the graph of y = x 2 of transformations (i.e., translations, reflec- tions in the x-axis, vertical stretches or compressions) by considering separately each parameter a, h, and k [i.e., investigate the effect on the graph of y = x2 of a,h, andkiny=x2 +k,y=(x–h)2,and
y = ax2];
– explain the roles of a, h, and k in
y = a(x – h )2 + k, using the appropriate terminology to describe the transforma- tions, and identify the vertex and the equa- tion of the axis of symmetry;
Relating the Graph of y
Transformations
x2 and Its