Linear Systems and Quadratics Overview

Determine the slope from A(-1,7) to B(2,6).

Given y = -2(x-5)(x + 3), state the direction of opening.

Given y = -2(x-5)(x + 3), state the zeros.

Given y = -2(x-5)(x + 3), state the equation of the axis of symmetry.

Given y = -2(x-5)(x + 3), calculate the coordinates of the vertex.

Two people are standing on the same side of a tree. The tree is 10 m tall. The angles of elevation from the people to the top of the tree are 25º and 30°. Determine the distance between the people.

A rocket is launched from the ground and follows a parabolic path. The rocket reached a maximum height of 50 m and returned to the ground 120 m from where it was launched. Algebraically determine the equation of its flight path. From the given information, you should be able to find the equation in factored form and in vertex form.

A photo measures 4cm by 6 cm. It is to be surrounded by a border for framing. The width of the border is the same on all sides of the photo. The area of the border is equal to the area of the photo. Calculate the dimensions of the frame that will be needed.

A plane has a flight path of 2x-5y = 15. Another plane has a flight path of 3x + 4y = 34. Determine where the planes intersect.

Two people are standing on the same side of a tree. The people are 10 m apart. The angles of elevation from the people to the top of the tree are 25° and 30°. Determine the height of the tree.

Does the ordered pair (2, 3) satisfy the equation 2x + 4y = 16? Show your work.

Solve the following linear system by graphing. Check your solution. y = 2x - 1 y = -x + 5

Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions. y = 2x - 5 4x - 2y = 10

Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions. 3x - y = 17 6x + 2y = -8

Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions. 12x + 8y + 4 = 0 15x + 10y = 5

Solve the following system of equations by substitution. Check your solution. 4x + 3y = 7 3x + y = -1

Solve the following system of equations by elimination. Check your solution: 3x + y = 17 2x - y = -2

Design (do not solve) a system of equations for the following scenarios: A supermarket sells 2-kg and 4-kg bags of sugar. A shipment of 1100 bags of sugar has a total mass of 2900 kg. How many 2-kg bags and 4-kg bags are in the shipment?

Design (do not solve) a system of equations for the following scenarios: The school car wash charged $5 for a car and $6 for a van. A total of 86 cars and vans were washed on Saturday, and the amount earned was $475. How many vans were washed on Saturday?

Determine the length of the line segment joining the points (3, 7) and (-1, -5). Round to the nearest tenth.

Find the slope of the line with points (0, 5) and (6, 10).

Write the equation for a circle with centre (0, 0) and through the point (3, 4).

The equation for a circle with centre (0, 0) is x² + y² = 361. What is the radius?

Determine the midpoint of the line segment with the endpoints (-6, 2) and (4, 8).

Explain how you can determine whether two lines are perpendicular or parallel based on their slopes.

The vertices of a quadrilateral are S(1,2), T(3, 5), U(6, 7), and V(4, 4). Verify that STUV is a parallelogram.

Find the equation of the median from vertex A in ΔABC, if the coordinates of the vertices are A(-3, -1), B(3, 5), and C(7, -3).

What is the degree of the following polynomials? 3x² - 2x

What is the degree of the following polynomials? 4a - b³

What is the degree of the following polynomials? 4x⁴ - 2x² + x² + 4

Expand and Simplify 2(m - 3)(m + 8)

Expand and Simplify (x + 4)²

Expand and Simplify 6(m - 2)(m + 3) - 3(3m - 4)

Factor the following completely 2ax + 10ay - 8az

Factor the following completely x² - 5x + 6

Factor the following completely 3x²y - 6x² - 2y + y²

Factor the following completely x² - 25

Factor the following completely 3y² + y - 4

Factor the following completely 4x² - 16x - 48

Phil wants to make the largest possible rectangular vegetable garden using 18 m of fencing. The garden is right behind the back of his house, so he has to fence it on only three sides. Determine the dimensions that maximize the area of the garden.

A pizza company's research shows that a $0.25 decrease in the price of a pizza results in 50 more pizzas being sold. The usual price of $15 for a three-item pizza results in sales of 1000 pizzas. Write the algebraic expression that models the maximum revenue for this situation, and find the price of a pizza that will produce a maximum revenue.

State the roots of each equation. a) (x-2)(x + 7) = 0

State the roots of each equation. b) (3x+1)(2x-3) = 0

Solve 3x²-6x-8 = 0 using the quadratic formula. Round to the nearest hundredth.

Use a calculator to find each of the following, to four decimal places. a) tan 84° =

Use a calculator to find each of the following, to four decimal places. c) cos 43° =

Find ∠K, to the nearest degree. a) tan ∠K = 2.750

Find ∠K, to the nearest degree. c) cos ∠K = 6/13

Solve each triangle. Round each side length to the nearest tenth of a unit, and each angle, to the nearest degree. a) S = 40°, T = 15 m, U = 19 m.

Solve each triangle. Round each side length to the nearest tenth of a unit, and each angle, to the nearest degree. b) W = 14 cm, X = 24 cm

From the window of one building, Sam finds the angle of depression of the top of a second building is 41° and the angle of depression of the bottom is 54°. The buildings are 56 m apart. Find, to the nearest metre, the height of the second building.

In ∆ABC, ∠A = 50°, a = 9 m, and b = 8 m. What is the measure of ∠B?

Linear System

Solving a Linear System

Solving by Graphing

Solving by Substitution

Solving by Elimination

Distance Formula

Slope

Equation of a Circle with Center (0, 0)

Midpoint Formula

Slopes of Parallel and Perpendicular Lines

Parallelogram

Median of a Triangle

Degree of a Polynomial

Expanding and Simplifying Polynomials

Factoring Polynomials

Parabola

Vertex of a Parabola

Axis of Symmetry

Maximum or Minimum Value

Vertex Form of a Quadratic Equation

Completing the Square

Roots of a Quadratic Equation

Quadratic Formula

Tangent (tan)

Sine (sin)

Cosine (cos)

Angle of Depression

Angle of Elevation

Law of Sines

Law of Cosines