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Questions and Answers
Determine the slope from A(-1,7) to B(2,6).
Determine the slope from A(-1,7) to B(2,6).
-1/3
Given y = -2(x-5)(x + 3), state the direction of opening.
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 zeros.
5, -3
Given y = -2(x-5)(x + 3), state the equation of the axis of symmetry.
Given y = -2(x-5)(x + 3), state the equation of the axis of symmetry.
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Given y = -2(x-5)(x + 3), calculate the coordinates of the vertex.
Given y = -2(x-5)(x + 3), calculate the coordinates of the vertex.
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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.
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.
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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 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.
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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 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.
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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.
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.
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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.
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.
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Does the ordered pair (2, 3) satisfy the equation 2x + 4y = 16? Show your work.
Does the ordered pair (2, 3) satisfy the equation 2x + 4y = 16? Show your work.
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Solve the following linear system by graphing. Check your solution. y = 2x - 1
y = -x + 5
Solve the following linear system by graphing. Check your solution. y = 2x - 1 y = -x + 5
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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. y = 2x - 5 4x - 2y = 10
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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. 3x - y = 17 6x + 2y = -8
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Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions.
12x + 8y + 4 = 0
15x + 10y = 5
Without graphing, determine whether each system has one solution, no solution, or infinitely many solutions. 12x + 8y + 4 = 0 15x + 10y = 5
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Solve the following system of equations by substitution. Check your solution. 4x + 3y = 7
3x + y = -1
Solve the following system of equations by substitution. Check your solution. 4x + 3y = 7 3x + y = -1
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Solve the following system of equations by elimination. Check your solution:
3x + y = 17
2x - y = -2
Solve the following system of equations by elimination. Check your solution: 3x + y = 17 2x - y = -2
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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: 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?
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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?
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?
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Determine the length of the line segment joining the points (3, 7) and (-1, -5). Round to the nearest tenth.
Determine the length of the line segment joining the points (3, 7) and (-1, -5). Round to the nearest tenth.
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Find the slope of the line with points (0, 5) and (6, 10).
Find the slope of the line with points (0, 5) and (6, 10).
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Write the equation for a circle with centre (0, 0) and through the point (3, 4).
Write the equation for a circle with centre (0, 0) and through the point (3, 4).
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The equation for a circle with centre (0, 0) is x² + y² = 361. What is the radius?
The equation for a circle with centre (0, 0) is x² + y² = 361. What is the radius?
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Determine the midpoint of the line segment with the endpoints (-6, 2) and (4, 8).
Determine the midpoint of the line segment with the endpoints (-6, 2) and (4, 8).
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Explain how you can determine whether two lines are perpendicular or parallel based on their slopes.
Explain how you can determine whether two lines are perpendicular or parallel based on their slopes.
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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.
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.
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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).
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).
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What is the degree of the following polynomials?
3x² - 2x
What is the degree of the following polynomials? 3x² - 2x
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What is the degree of the following polynomials?
4a - b³
What is the degree of the following polynomials? 4a - b³
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What is the degree of the following polynomials?
4x⁴ - 2x² + x² + 4
What is the degree of the following polynomials? 4x⁴ - 2x² + x² + 4
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Expand and Simplify
2(m - 3)(m + 8)
Expand and Simplify 2(m - 3)(m + 8)
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Expand and Simplify
(x + 4)²
Expand and Simplify (x + 4)²
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Expand and Simplify
6(m - 2)(m + 3) - 3(3m - 4)
Expand and Simplify 6(m - 2)(m + 3) - 3(3m - 4)
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Factor the following completely
2ax + 10ay - 8az
Factor the following completely 2ax + 10ay - 8az
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Factor the following completely
x² - 5x + 6
Factor the following completely x² - 5x + 6
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Factor the following completely
3x²y - 6x² - 2y + y²
Factor the following completely 3x²y - 6x² - 2y + y²
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Factor the following completely
x² - 25
Factor the following completely x² - 25
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Factor the following completely
3y² + y - 4
Factor the following completely 3y² + y - 4
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Factor the following completely
4x² - 16x - 48
Factor the following completely 4x² - 16x - 48
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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.
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.
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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.
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.
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State the roots of each equation. a) (x-2)(x + 7) = 0
State the roots of each equation. a) (x-2)(x + 7) = 0
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State the roots of each equation. b) (3x+1)(2x-3) = 0
State the roots of each equation. b) (3x+1)(2x-3) = 0
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Solve 3x²-6x-8 = 0 using the quadratic formula. Round to the nearest hundredth.
Solve 3x²-6x-8 = 0 using the quadratic formula. Round to the nearest hundredth.
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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. a) tan 84° =
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Use a calculator to find each of the following, to four decimal places. c) cos 43° =
Use a calculator to find each of the following, to four decimal places. c) cos 43° =
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Find ∠K, to the nearest degree. a) tan ∠K = 2.750
Find ∠K, to the nearest degree. a) tan ∠K = 2.750
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Find ∠K, to the nearest degree. c) cos ∠K = 6/13
Find ∠K, to the nearest degree. c) cos ∠K = 6/13
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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. a) S = 40°, T = 15 m, U = 19 m.
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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
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
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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.
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.
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In ∆ABC, ∠A = 50°, a = 9 m, and b = 8 m. What is the measure of ∠B?
In ∆ABC, ∠A = 50°, a = 9 m, and b = 8 m. What is the measure of ∠B?
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Study Notes
Linear Systems
- A linear system with two equations can have one solution, no solution, or infinitely many solutions.
- If the slopes of the lines are different, there is one solution where the two lines intersect.
- If the slopes are the same and the y-intercepts are different, there is no solution; the lines do not intersect.
- If the slopes and y-intercepts are the same, there are infinitely many solutions; the lines are the same.
Analytic Geometry
- The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals of each other.
Quadratics
- The degree of a polynomial is the highest power of the variable in the polynomial.
- Polynomials can be expanded and simplified using algebraic methods.
- Factoring can simplify polynomials, which helps find the zeros.
- Quadratic functions can be graphed and analyzed to find the vertex, axis of symmetry, and intercepts.
Trigonometry
- Trigonometric functions (sine, cosine, tangent) relate angles and sides of right triangles.
- Trigonometric ratios are used to determine lengths and angles in triangles.
Coordinate Geometry
- The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ - x₁)² + (y₂ - y₁)²).
- The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is (y₂ - y₁)/(x₂ - x₁).
- The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
Problem Solving
- Problem-solving strategies can be applied to solve various mathematical problems.
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Description
Test your understanding of linear systems and quadratic functions with this quiz. It covers concepts such as the solutions of linear equations, properties of lines in analytic geometry, and the analysis of quadratic functions. Get ready to explore the relationships between these foundational algebra topics!
