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Overall Expectation (Analytic Geometry):
Model and solve problems involving the intersection of two straight lines
New Specific Expectations (From Addendum Winter 2022):
Identify the relationship between the slopes of parallel and perpendicular lines, and use this relationship to solve related problems;
Develop the formula for the slope of a line (i.e., ), and use this formula to determine the equations of lines, given information about the lines (e.g., a graph of a line, a table of values, the coordinates of two points);
Represent the equations of lines in different forms (e.g., y = mx + b, Ax + By + C = 0, Ax + By = D) and translate between these forms, as appropriate for the context.
Overall Expectation (Analytic Geometry):
Model and solve problems involving the intersection of two straight lines
Specific Expectations:
Solve systems of two linear equations involving two variables, using the algebraic methods of substitution or elimination
Solve problems that arise from realistic situations described in words that are represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method
Overall Expectation :
Solve quadratic equations and interpret the solutions with respect to the corresponding relations
Specific Expectations:
Expand and simplify second-degree polynomial expressions, using a variety of tools and strategies
Factor polynomial expressions involving common factors, trinomials, and difference of squares, using a variety of tools and strategies
Overall Expectation:
Determine the basic properties of quadratic relations (Part 1)
Relate transformations of the graph y = x^2 to the algebraic representation y = a(x - h)^2 + k
Solve quadratic equations and interpret the solutions with respect to the corresponding relations
Specific Expectations:
(Part 1)
Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology, or from secondary sources; graph the data and draw a curve of best fit, if appropriate, with or without the use of technology
Determine, through investigation with and without the use of technology, that a quadratic relation of the form y = ax^2 + bx + c (a not 0) can be graphically represented as a parabola, and that the table of values yields a constant second difference
Identify the key features of a graph of a parabola, and use the appropriate terminology to describe them
Compare, through technology, the features of the graph of y = x^2 and the graph of y = 2^x, and determine the meaning of a negative exponent and of zero as an exponent
(Part 2)
Identify, through investigation using technology, the effect on the graph of y = x^2 of transformations by considering separately each parament a, h, and k
Explain the roles of a, h, and k in y = a(x - h)^2 + k, using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of symmetry
Sketch, by hand, the graph of y = a(x - h)^2 + k by applying transformations to the graph of y = x2
Determine the equation, in the form y = a(x - h)^2 + k, opf a given graph of a parabola
(Part 3)
Determine through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic reaction, expressions in the form y = a(x - r)(x - s)
Overall Expectation:
Solve quadratic equations and interpret the solutions with respect to the corresponding relations
Solve problems involving quadratic relations
Specific Expectations:
Interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corresponding relations
Express y = ax2 + bx + c in the form y = a(x - h)2 + k by completing the square in situations involving no fractions, using a variety of tools
Sketch or graph a quadratic relation whose equation is given in the form y = ax2 + bc + c, using a variety of methods
Explore the algebraic development of the quadratic formula
Solve quadratic equations that have real roots, using a variety of methods
Overall Expectation (Analytic Geometry):
Using their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity (Part 1)
Solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem (Part 2)
Solve problems involving acute triangles, using the sine law and the cosine law (Part 3)
Specific Expectations:
(Part 1)
Verify, through investigation, the properties of similar triangles
Describe and compare the concepts of similarity and congruence
Solve problems involving similar triangles in realistic situations
(Part 2)
Determine, through investigation, the relationship between the ratio of two sides of a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios
Determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem
Solve problems involving the measures of sides and angles in right triangles in real-life applications, using the primary trigonometric ratios and the Pythagorean theorem
(Part 3)
Explore the development of the sine law within acute triangles*
Explore the development of the cosine law within acute triangles*
Determine the measures of sides and angles in acute triangles, using the sine law and cosine law
Solve problems involving the measures of sides and angles of acute triangles
* Student reproduction of the development of the sine and cosine law formula is not required
Overall Expectation (Analytic Geometry):
Solve problems using analytical geometry involving properties of lines and line segments
Verify geometric properties of triangles and quadrilaterals, using analytic geometry
Specific Expectations:
Develop the formula for the midpoint of a line segment, and use this formula to solve problems
Development the formula for the length of a line segment, and use this formula to solve problems
Development the equation for a circle with centre (0,0) and radius r, by applying the formula for the length of a line segment
Determine the radius of a circle with centre (0,0), given its equation; write the equation of a circle with the centre (0,0), given the radius; and sketch the circle, given the equation in the form x2 + y2 = r2
Solve problems involving the slope, length, and midpoint of a line segment
Determine, through investigation, some characteristics of geometric figures
Verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures
Plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property
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