3 Lecture

1. What is the equation of the vertical line passing through the point (-3,5)? a) x = -3 b) y = -3 c) x = 5 d) y = 5

Solution: a) x = -3

2. What are the coordinates of the origin on a coordinate plane? a) (1,1) b) (-1,-1) c) (0,0) d) (2,2)

Solution: c) (0,0)

3. What is the slope of the line passing through the points (3,5) and (1,2)? a) 3/2 b) -3/2 c) 2/3 d) -2/3

Solution: b) -3/2

4. Which quadrant contains the point (-4,-2)? a) First b) Second c) Third d) Fourth

Solution: c) Third

5. What is the distance between points (2,5) and (-3,1)? a) 2 b) 5 c) sqrt(26) d) sqrt(29)

Solution: d) sqrt(29)

6. What is the slope of the line perpendicular to the line y = 3x - 2? a) 3/2 b) -3/2 c) -1/3 d) 1/3

Solution: c) -1/3

7. Which of the following is an equation of a vertical line? a) y = 2x + 3 b) x = 4 c) y = -x + 1 d) x + y = 7

Solution: b) x = 4

8. What is the equation of the line passing through the points (2,-3) and (4,5)? a) y = -2x + 1 b) y = 2x - 7 c) y = -4x - 11 d) y = 4x - 11

Solution: d) y = 4x - 11

9. What is the slope-intercept form of the equation of the line passing through the point (2,4) with a slope of -2? a) y = -2x - 4 b) y = -2x + 8 c) y = 2x - 4 d) y = 2x + 4

Solution: a) y = -2x + 8

10. What is the equation of the line passing through the points (-1,3) and (5,-1)? a) y = -x + 2 b) y = x + 2 c) y = -x - 2 d) y = x - 2

Solution: c) y = -x + 2

1. What is a coordinate plane? Answer: A coordinate plane is a two-dimensional plane that is divided into four quadrants, labeled I, II, III, and IV. The plane is defined by two perpendicular axes, the x-axis and the y-axis, which intersect at the origin, denoted as (0,0).

   
   

2. What is the x-axis? Answer: The x-axis is the horizontal axis on a coordinate plane.

   
   

3. What is the y-axis? Answer: The y-axis is the vertical axis on a coordinate plane.

   
   

4. What is the origin? Answer: The origin is the point (0,0) on a coordinate plane where the x-axis and the y-axis intersect.

   
   

5. What is the slope of a line? Answer: The slope of a line is the ratio of the change in the y-coordinate to the change in the x-coordinate.

   
   

6. What is the y-intercept? Answer: The y-intercept is the point where a line intersects the y-axis.

   
   

7. What is a linear equation? Answer: A linear equation is an equation that can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

   
   

8. What is a quadratic equation? Answer: A quadratic equation is an equation that can be written in the form y = ax^2 + bx + c, where a, b, and c are constants.

   
   

9. What is a parabola? Answer: A parabola is a U-shaped curve that is the graph of a quadratic equation.

   
   

10. How do you find the vertex of a parabola? Answer: The vertex of a parabola can be found by using the formula x = -b/2a to find the x-coordinate, and then plugging that value into the quadratic equation to find the corresponding y-coordinate.

Coordinate Planes and Graphs

Coordinate planes and graphs are fundamental concepts in calculus and analytical geometry. They provide a way to visually represent relationships between mathematical quantities, making it easier to understand complex concepts. A coordinate plane is a two-dimensional plane that is divided into four quadrants, labeled I, II, III, and IV. The plane is defined by two perpendicular axes, the x-axis, and the y-axis, which intersect at the origin, denoted as (0,0). The x-axis is the horizontal axis, while the y-axis is the vertical axis. The two axes divide the plane into four quadrants, with quadrant I in the upper right-hand corner, quadrant II in the upper left-hand corner, quadrant III in the lower left-hand corner, and quadrant IV in the lower right-hand corner. To graph a point on the coordinate plane, we need to specify its x-coordinate and its y-coordinate. The x-coordinate represents the point's position along the x-axis, while the y-coordinate represents its position along the y-axis. The point is then located at the intersection of the x-coordinate and the y-coordinate. For example, point (2,3) is located two units to the right of the origin and three units above the origin. Graphing a linear equation is another important concept in calculus and analytical geometry. A linear equation can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is the ratio of the change in the y-coordinate to the change in the x-coordinate. The y-intercept is the point where the line intersects the y-axis. To graph a linear equation, we can first plot the y-intercept on the y-axis. We can then use the slope to find additional points on the line. If the slope is positive, we can move up and to the right to find additional points, while if the slope is negative, we can move down and to the right. Once we have found two or more points on the line, we can draw a straight line through them to graph the equation. Graphing a quadratic equation is another important concept in calculus and analytical geometry. A quadratic equation can be written in the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The direction of the opening of the parabola depends on the sign of the coefficient a. To graph a quadratic equation, we can first find the vertex of the parabola, which is the point where the parabola changes direction. The x-coordinate of the vertex is given by -b/2a, while the y-coordinate is given by the value of the function at the vertex. We can then find additional points on the parabola by plugging in other values of x into the equation. Once we have found enough points, we can draw the parabola by connecting them with a smooth curve. In conclusion, coordinate planes and graphs are essential tools in calculus and analytical geometry. They provide a visual representation of mathematical concepts, making it easier to understand complex relationships between mathematical quantities. Graphing linear and quadratic equations is an important skill that is used in a variety of fields, including engineering, physics, and economics.