The Coordinate Plane
1) terms: origin, x-axis, y-axis, x-coordinate, y-coordinate
2) every point in the plane can be denoted by a unique ordered pair, (x,y).
3) the axes divide the plane into four quadrants; the quadrant of a point determines the +/- sign of its x- and y-coordinates
The x-y plane/the rectangular coordinate system/the Cartesian plane
Graphing Lines
1) every line in the x-y plane has its own unique equation
2) every point on the line satisfies the equation of the line
3) we can figure out the graph of a line by plotting individual points. This is one option, and we will learn other options later
Vertical and Horizontal Lines
1) horizontal lines have the general form y = k
2) vertical lines have the general form x = k
3) the x-axis is y=0. The y-axis is x=0
4) if two points share the same x-coordinate, they are vertically separated
5) if two points share the same y-coordinate, they are horizontally separated
Slope
1) slope is rise over run (run-horizontal; rise-vertical)
2) we find the slope btw two points with a slope triangle, or with the slope formula
3) the slope of a line is the slope btw any two points on the line
4) lines with slopes of +/-1 make 45 degree with the axes
5) parallel lines have equal slopes (m_1 = m_2)
6) perpendicular lines have slopes that are opposite-signed reciprocals of each other: m_1 = -1/m_2
Intercepts
Vocabulary: Parabola 抛物线
An intercept is the point at which a graph crosses the x or y axis
At a y-intercept, the value of x = 0
At a x-intercept, the value of y = 0
Slope - Intercept Form
If we solve the equation of any line of y, we automatically put the equation into slope-intercept form, y=mx + b, where m is the slope and b is the y-intercept. This form makes it easy to graph the line and to understand where it goes
Writing Equations of Lines
1) if we are given a well labeled graph, we may be able to read the slope&y-intercept from the graph itself
2) if we are given two points, we can find a slope. Once we have a point and a slope, we can plug these into y = Mx + b to solve for b.
3) remember that you often have the option of solving algebraically or graphically, thinking about the proportions involved with slope.
Distance Between Two Points
1) horizontal & vertical distances -> subtraction
2) for slanted distances, draw (or imagine) the slope triangle, and use the Pythagorean theorem
3) remember Pythagorean triplets. Remember to use scale factors to simplify calculations
4) for a circle in the x-y plane, each slanted radius for a slope triangle with an equal hypotenuse.
5) the equation of a circle with radius r centered at (0,0) is x^2 + y^2 = r^2
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Reflection in the x-y Plane
1) reflection over x-axis -> same x, opposite +/- y.
2) reflection over y-axis -> same y, opposite +/- x.
3) reflection over y = x -> switch x & y
4) reflection over y = -x -> switch x & y and make each the opposite +/- sign
5) the mirror line is always the perpendicular bisector of the segment between the original point and its reflected image
6) any point on the mirror line is equidistant from the original point and its reflected image
Graphs of Quadratics
1) the graph of a quadratic二元方程式 is a parabola抛物线
2) if a is the quadratic coefficient, then a > 0 means the parabola opens upward; when a < 0, it opens downward
3)if |a|>1, then the parabola is skinny; if |a|<1, then the parabola is wide
4) the x-intercepts of the graph are the solutions of the quadratic set equal to zero
