Lesson 4 Part 1
Coordinates are the numbers used to give positions of points in a plane or space. In two dimensions, the coordinate plane has two number lines called the x-axis and the y-axis. The x-axis runs horizontally, and the y-axis runs vertically. The point where the two axes cross is called the origin.
For example, Point (4, 3) represents 4 as its x-coordinate, and 3 as its y-coordinate. The x-coordinate and y-coordinate respectively tell the signed distance from the y-axis and the x-axis. If the point is to the left of the origin, its x-coordinate is negative, and if the point is below the origin, its y-coordinate is negative.
For example, Point (4, 3) represents 4 as its x-coordinate, and 3 as its y-coordinate. The x-coordinate and y-coordinate respectively tell the signed distance from the y-axis and the x-axis. If the point is to the left of the origin, its x-coordinate is negative, and if the point is below the origin, its y-coordinate is negative.
Lesson 4 Part 1-1
Coordinates are the numbers used to give positions of points in a plane or space.
Lesson 4 Part 1-2
In two dimensions, the coordinate plane has two number lines called the x-axis and the y-axis.
Lesson 4 Part 1-3
The x-axis runs horizontally, and the y-axis runs vertically.
Lesson 4 Part 1-4
The point where the two axes cross is called the origin.
Lesson 4 Part 1-5
For example, Point (4, 3) represents 4 as its x-coordinate, and 3 as its y-coordinate.
Lesson 4 Part 1-6
The x-coordinate and y-coordinate respectively tell the signed distance from the y-axis and the x-axis.
Lesson 4 Part 1-7
If the point is to the left of the origin, its x-coordinate is negative.
Lesson 4 Part 1-8
And if the point is below the origin, its y-coordinate is negative.
Lesson 4 Part 2
Graphs are a good way to show formulae in visual form. Functions like y = 2x – 3 and y = -x + 3 can be graphed like those in Figure 1. This type of function is called a linear function because the graphs of such functions are drawn as straight lines. In the graph of y = 2x – 3, the slope of it is 2, and the y-intercept is -3.
When an object is moving in a straight line at a constant speed, the object’s travel distance increases in proportion to the time. The graph in Figure 2 shows the relationship between the distance and the time. The x-axis represents the time, and the y-axis represents the distance. The straight line that crosses two axes at the origin shows a proportional relationship. The gradient of the line represents the velocity of the object.
When an object is moving in a straight line at a constant speed, the object’s travel distance increases in proportion to the time. The graph in Figure 2 shows the relationship between the distance and the time. The x-axis represents the time, and the y-axis represents the distance. The straight line that crosses two axes at the origin shows a proportional relationship. The gradient of the line represents the velocity of the object.
Lesson 4 Part 2-1
Graphs are a good way to show formulae in visual form.
Lesson 4 Part 2-2
Functions like y = 2x – 3 and y = -x + 3 can be graphed like those in Figure 1.
Lesson 4 Part 2-3
This type of function is called a linear function because the graphs of such functions are drawn as straight lines.
Lesson 4 Part 2-4
In the graph of y = 2x – 3, the slope of it is 2.
Lesson 4 Part 2-5
And the y-intercept is -3.
Lesson 4 Part 2-6
When an object is moving in a straight line at a constant speed.
Lesson 4 Part 2-7
The object’s travel distance increases in proportion to the time.
Lesson 4 Part 2-8
The graph in Figure 2 shows the relationship between the distance and the time.
Lesson 4 Part 2-9
The x-axis represents the time, and the y-axis represents the distance.
Lesson 4 Part 2-10
The straight line that crosses two axes at the origin shows a proportional relationship.
Lesson 4 Part 2-11
The gradient of the line represents the velocity of the object.
Lesson 4 Part 3
The curve in Figure 1 shows a quadratic function y = x2. A quadratic function is generally described by the form y = ax2 + bx + c.
The shape of the curve for a quadratic function is a parabola. In general, a parabola will open upward and have a minimum point when the coefficient of x2 is positive. It will open downward and have a maximum point when the coefficient of x2 is negative.
To solve a quadratic equation x2 = 9, you have to find the numbers whose square is 9. The answers are 3 and -3, because both 32 and (-3)2 are 9.
However, when you solve an equation like x2 = 5, you cannot find its solutions in integers or decimals. In this case, you can express the answer ±√5, which is the number whose square is 5.
The shape of the curve for a quadratic function is a parabola. In general, a parabola will open upward and have a minimum point when the coefficient of x2 is positive. It will open downward and have a maximum point when the coefficient of x2 is negative.
To solve a quadratic equation x2 = 9, you have to find the numbers whose square is 9. The answers are 3 and -3, because both 32 and (-3)2 are 9.
However, when you solve an equation like x2 = 5, you cannot find its solutions in integers or decimals. In this case, you can express the answer ±√5, which is the number whose square is 5.
Lesson 4 Part 3-1
The curve in Figure 1 shows a quadratic function y = x2.
Lesson 4 Part 3-2
A quadratic function is generally described by the form.
Lesson 4 Part 3-3
y = ax2 + bx + c.
Lesson 4 Part 3-4
The shape of the curve for a quadratic function is a parabola.
Lesson 4 Part 3-5
In general, a parabola will open upward and have a minimum point when the coefficient of x2 is positive.
Lesson 4 Part 3-6
It will open downward and have a maximum point when the coefficient of x2 is negative.
Lesson 4 Part 3-7
To solve a quadratic equation x2 = 9.
Lesson 4 Part 3-8
You have to find the numbers whose square is 9.
Lesson 4 Part 3-9
The answers are 3 and -3.
Lesson 4 Part 3-10
Because both 32 and (-3)2 are 9.
Lesson 4 Part 3-11
However, when you solve an equation like x2 = 5.
Lesson 4 Part 3-12
You cannot find its solutions in integers or decimals.
Lesson 4 Part 3-13
In this case, you can express the answer ±√5.
Lesson 4 Part 3-14
Which is the number whose square is 5.