Coordinate geometry
61 flashcards covering Coordinate geometry for the SAT Math section.
Coordinate geometry is the branch of math that deals with plotting and analyzing shapes, lines, and points on a graph using coordinates, like (x, y) values. It helps you visualize relationships between objects in a two-dimensional plane, such as finding the distance between two points or the slope of a line. This topic builds on basic algebra and geometry, making it essential for understanding how equations translate into visual forms, which is useful in real-world applications like mapping or physics.
On the SAT Math section, coordinate geometry appears in questions that test your ability to graph lines and curves, calculate distances or midpoints, and solve for equations of lines and circles. Common traps include mixing up positive and negative slopes or misreading graph scales, so watch for details in word problems. Focus on mastering formulas like the distance formula and slope-intercept form, as these often lead to straightforward multiple-choice answers. A solid grasp here can boost your score by improving accuracy on about 10-15% of the questions.
Practice sketching graphs quickly to catch errors early.
Terms (61)
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Coordinate Plane
The coordinate plane is a two-dimensional surface formed by two perpendicular number lines called axes, used to plot points and graph equations.
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X-axis
The x-axis is the horizontal number line on the coordinate plane, where the y-coordinate of any point on it is zero.
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Y-axis
The y-axis is the vertical number line on the coordinate plane, where the x-coordinate of any point on it is zero.
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Quadrants
Quadrants are the four regions on the coordinate plane divided by the axes, numbered counterclockwise from the top right, with signs for coordinates as follows: first quadrant positive, second negative x and positive y, third negative, and fourth positive x and negative y.
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Ordered Pair
An ordered pair is a pair of numbers (x, y) that represents a point's location on the coordinate plane, with x indicating the horizontal distance from the origin and y the vertical.
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Distance Formula
The distance formula calculates the straight-line distance between two points (x1, y1) and (x2, y2) as sqrt((x2 - x1)^2 + (y2 - y1)^2).
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Midpoint Formula
The midpoint formula finds the coordinates of the midpoint between two points (x1, y1) and (x2, y2) as ((x1 + x2)/2, (y1 + y2)/2).
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Slope
Slope measures the steepness of a line and is calculated as the change in y-coordinates divided by the change in x-coordinates between any two points on the line.
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Slope-Intercept Form
Slope-intercept form is the equation of a line written as y = mx + b, where m is the slope and b is the y-intercept.
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Point-Slope Form
Point-slope form is the equation of a line written as y - y1 = m(x - x1), using a known point (x1, y1) and slope m.
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Standard Form of a Line
Standard form of a line is the equation written as Ax + By = C, where A, B, and C are constants, and A and B are not both zero.
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Parallel Lines
Parallel lines on the coordinate plane have the same slope but different y-intercepts, meaning they never intersect.
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Perpendicular Lines
Perpendicular lines on the coordinate plane have slopes that are negative reciprocals of each other, forming a 90-degree angle where they intersect.
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X-intercept
The x-intercept is the point where a line crosses the x-axis, meaning the y-coordinate is zero.
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Y-intercept
The y-intercept is the point where a line crosses the y-axis, meaning the x-coordinate is zero.
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Horizontal Line
A horizontal line has a slope of zero and an equation of the form y = k, where k is a constant, extending infinitely left and right.
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Vertical Line
A vertical line has an undefined slope and an equation of the form x = h, where h is a constant, extending infinitely up and down.
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Equation of a Circle
The equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.
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Center of a Circle
The center of a circle is the point (h, k) in its equation, equidistant from all points on the circle.
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Radius
The radius is the distance from the center of a circle to any point on its circumference.
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Parabola
A parabola is a U-shaped curve that is the graph of a quadratic equation, such as y = ax^2 + bx + c.
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Vertex of a Parabola
The vertex of a parabola is its highest or lowest point, found at x = -b/(2a) for the equation y = ax^2 + bx + c.
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Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex of a parabola, dividing it into two mirror-image halves.
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Vertex Form of a Parabola
Vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex.
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Graphing Linear Equations
Graphing linear equations involves plotting points that satisfy the equation and drawing a straight line through them.
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Graphing Inequalities
Graphing inequalities involves shading the region on the coordinate plane that satisfies the inequality, such as y > mx + b.
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Boundary Line
The boundary line in a graph of an inequality is the line that represents the equality, such as y = mx + b for y > mx + b.
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Shading Region
The shading region on a graph of an inequality indicates the set of points that satisfy the inequality, such as above the line for y > mx + b.
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Systems of Equations
Systems of equations involve finding the point of intersection of two or more equations, often by solving them simultaneously.
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Point of Intersection
The point of intersection is where two lines cross on the coordinate plane, satisfying both equations simultaneously.
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Distance from a Point to a Line
The distance from a point to a line is the shortest perpendicular distance, calculated using the formula for the line in standard form.
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Reflection over the X-axis
Reflection over the x-axis changes the sign of the y-coordinate of a point, such as (x, y) becoming (x, -y).
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Reflection over the Y-axis
Reflection over the y-axis changes the sign of the x-coordinate of a point, such as (x, y) becoming (-x, y).
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Reflection over Y = X
Reflection over the line y = x swaps the x and y coordinates of a point, such as (x, y) becoming (y, x).
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Translation
Translation moves every point of a graph by a fixed amount in the x and y directions, such as shifting right by h and up by k.
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Positive Slope
A positive slope indicates that the line rises from left to right, meaning as x increases, y increases.
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Negative Slope
A negative slope indicates that the line falls from left to right, meaning as x increases, y decreases.
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Zero Slope
Zero slope means the line is horizontal, with no change in y as x varies.
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Undefined Slope
Undefined slope occurs in vertical lines, where x is constant and y varies without bound.
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Linear Function
A linear function has a graph that is a straight line, represented by an equation of the form y = mx + b.
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Non-Linear Function
A non-linear function has a graph that is not a straight line, such as a parabola or circle.
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Domain on Graphs
Domain on graphs refers to the set of all possible x-values, often determined by the leftmost and rightmost points.
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Range on Graphs
Range on graphs refers to the set of all possible y-values, often determined by the lowest and highest points.
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Symmetry in Graphs
Symmetry in graphs means one half is a mirror image of the other, such as even functions symmetric about the y-axis.
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Area of a Triangle on Plane
The area of a triangle on the coordinate plane can be found using the formula (1/2)|(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))| for vertices (x1,y1), (x2,y2), (x3,y3).
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Perimeter on Coordinate Plane
Perimeter on the coordinate plane is the total distance around a shape, calculated by summing the distances between its vertices.
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Pythagorean Theorem in Coordinates
The Pythagorean theorem in coordinates is used in the distance formula, relating the distances of the sides of a right triangle.
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Common Trap: Slope Calculation
A common trap in slope calculation is dividing incorrectly or confusing the order of points, leading to sign errors.
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Common Trap: Vertical Lines
A common trap with vertical lines is trying to find a slope, which is undefined, often resulting in division by zero errors.
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Rate of Change
Rate of change in coordinate geometry is the slope of a line, representing how much y changes per unit change in x.
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Initial Value
Initial value in a linear function is the y-intercept, representing the starting point when x is zero.
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Intercept Form
Intercept form of a line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.
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Feasible Region
The feasible region is the area on the coordinate plane that satisfies all inequalities in a system, often used in optimization problems.
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Break-Even Point
The break-even point is the intersection of cost and revenue lines on the coordinate plane, where neither profit nor loss occurs.
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Absolute Value Function
An absolute value function, like y = |x|, graphs as a V-shape with the vertex at the origin, reflecting parts of the line over the x-axis.
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Strategy for Finding Intercepts
A strategy for finding intercepts is to set y to zero for the x-intercept and x to zero for the y-intercept, then solve the equation.
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Example: Distance Between Points
For points (1, 2) and (4, 6), the distance is sqrt((4-1)^2 + (6-2)^2) = sqrt(18) = 3sqrt(2).
Points (1, 2) and (4, 6) yield a distance of about 4.24 units.
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Example: Midpoint of Segment
For endpoints (2, 3) and (6, 7), the midpoint is ((2+6)/2, (3+7)/2) = (4, 5).
Segment from (2, 3) to (6, 7) has midpoint at (4, 5).
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Example: Equation from Two Points
For points (1, 2) and (3, 6), the slope is (6-2)/(3-1) = 2, so using point-slope, y - 2 = 2(x - 1), or y = 2x.
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Example: Parallel Line Equation
If a line is y = 3x + 2, a parallel line through (1, 1) has slope 3, so y - 1 = 3(x - 1), or y = 3x - 2.
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Example: Circle with Given Center
A circle with center (2, 3) and radius 4 has the equation (x - 2)^2 + (y - 3)^2 = 16.