Coordinate geometry
51 flashcards covering Coordinate geometry for the ACT Math section.
Coordinate geometry is a way to describe shapes and points using a grid called the coordinate plane, where each point has an x and y value. It helps us visualize and calculate relationships between objects, like the distance between two points or the slope of a line. This topic is essential because it bridges algebra and geometry, allowing you to solve real-world problems more precisely.
On the ACT Math section, coordinate geometry shows up in questions about graphing lines, finding equations, calculating distances or midpoints, and working with circles or parabolas. Common traps include algebraic errors, such as forgetting to square terms in the distance formula, or misinterpreting graph scales. Focus on practicing quick graphing, understanding slope-intercept form, and double-checking calculations to avoid these pitfalls.
A concrete tip: Always plot points on paper to visualize problems accurately.
Terms (51)
- 01
Coordinate plane
The coordinate plane is a two-dimensional surface formed by two perpendicular number lines called the x-axis and y-axis, which intersect at the origin and are used to plot points and graph equations.
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Ordered pair
An ordered pair is a pair of numbers (x, y) that represents a point on the coordinate plane, where x is the horizontal distance from the origin and y is the vertical distance.
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X-axis
The x-axis is the horizontal number line on the coordinate plane, where points have a y-coordinate of zero and are used to measure horizontal positions.
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Y-axis
The y-axis is the vertical number line on the coordinate plane, where points have an x-coordinate of zero and are used to measure vertical positions.
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Origin
The origin is the point (0, 0) on the coordinate plane where the x-axis and y-axis intersect.
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Quadrants
Quadrants are the four regions on the coordinate plane divided by the x-axis and y-axis: Quadrant I (x > 0, y > 0), Quadrant II (x < 0, y > 0), Quadrant III (x < 0, y < 0), and Quadrant IV (x > 0, y < 0).
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Distance formula
The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as sqrt((x2 - x1)^2 + (y2 - y1)^2), which is derived from the Pythagorean theorem.
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Midpoint formula
The midpoint formula finds the midpoint of a line segment between two points (x1, y1) and (x2, y2) as ((x1 + x2)/2, (y1 + y2)/2), representing the average of the coordinates.
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Slope
Slope is a measure of the steepness of a line, calculated as the change in y-coordinates divided by the change in x-coordinates between two points on the line, often denoted as m.
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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, making it easy to graph by identifying the slope and initial point.
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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, which is useful for lines passing through a specific point.
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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 it can be used to easily find x- and y-intercepts by setting the other variable to zero.
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Parallel lines
Parallel lines are lines in the same plane that never intersect and have the same slope, meaning their equations will have identical m values in slope-intercept form.
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Perpendicular lines
Perpendicular lines are lines that intersect at a right angle, and their slopes are negative reciprocals of each other, such as if one has slope 2, the other has slope -1/2.
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Equation of a circle
The equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius, representing all points equidistant from the center.
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X-intercept
The x-intercept is the point where a graph crosses the x-axis, meaning the y-coordinate is zero, and it can be found by setting y = 0 in the equation and solving for x.
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Y-intercept
The y-intercept is the point where a graph crosses the y-axis, meaning the x-coordinate is zero, and it can be found by setting x = 0 in the equation and solving for y.
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Horizontal line
A horizontal line has a constant y-value and a slope of zero, represented by an equation like y = k, where k is the y-intercept, and it extends infinitely left and right.
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Vertical line
A vertical line has a constant x-value and an undefined slope, represented by an equation like x = h, where h is the x-intercept, and it extends infinitely up and down.
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Undefined slope
Undefined slope occurs in vertical lines where the change in x is zero, making the slope calculation impossible as division by zero, indicating a line parallel to the y-axis.
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Zero slope
Zero slope occurs in horizontal lines where the change in y is zero, meaning the line is flat and parallel to the x-axis, as in y = constant.
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Graphing a linear equation
Graphing a linear equation involves plotting at least two points that satisfy the equation and drawing a straight line through them, using forms like slope-intercept for ease.
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Systems of linear equations
Systems of linear equations involve two or more equations graphed on the same plane, with the solution being the point of intersection where both equations are satisfied.
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Intersection point
The intersection point is where two or more graphs cross, representing the solution to a system of equations by solving them simultaneously.
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Parabola
A parabola is a U-shaped graph of a quadratic equation, such as y = ax^2 + bx + c, which opens upward if a > 0 or downward if a < 0.
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Vertex of a parabola
The vertex of a parabola is the highest or lowest point, found at x = -b/(2a) for y = ax^2 + bx + c, representing the maximum or minimum value.
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Axis of symmetry
The axis of symmetry is the vertical line that passes through the vertex of a parabola, given by x = -b/(2a) for a quadratic equation, dividing the graph into mirror images.
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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, allowing easy identification of the vertex and direction of the graph.
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Translations of graphs
Translations of graphs involve shifting a graph horizontally or vertically, such as adding a constant to x or y in the equation, like y = f(x - h) + k shifts right by h and up by k.
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Reflections of graphs
Reflections of graphs involve flipping a graph over an axis, such as multiplying x by -1 for reflection over the y-axis or y by -1 for reflection over the x-axis.
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Slope as rate of change
Slope represents the rate of change in a linear relationship, indicating how much y changes for each unit increase in x, often used in real-world contexts like speed.
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Equation from two points
To find the equation from two points, first calculate the slope using the two points, then use point-slope form with one of the points to derive the full equation.
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Common trap: Confusing slope and intercept
A common trap is mixing up slope and y-intercept in equations, such as thinking b in y = mx + b is the slope instead of the y-value where the line crosses the y-axis.
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Graphing inequalities
Graphing inequalities involves plotting the boundary line and shading the appropriate region, using a solid line for ≤ or ≥ and a dashed line for < or >, based on the inequality sign.
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Shading regions for inequalities
Shading regions for inequalities means filling the area on the graph that satisfies the inequality, such as above the line for y > mx + b or below for y < mx + b.
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Absolute value graph
The graph of an absolute value equation like y = |x| is a V-shaped figure with the vertex at the origin, reflecting the positive and negative parts of the function.
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Piecewise functions
Piecewise functions are defined by different expressions over different intervals, graphed by plotting each piece separately and connecting as needed based on the domain.
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Domain from a graph
Domain from a graph is the set of all x-values shown on the graph, typically from the leftmost to rightmost points, indicating where the function is defined.
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Range from a graph
Range from a graph is the set of all y-values shown, from the lowest to highest points, representing the output values of the function.
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Symmetry about x-axis
Symmetry about the x-axis means that if a point (x, y) is on the graph, then (x, -y) is also on the graph, as in even functions like y = x^2.
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Symmetry about y-axis
Symmetry about the y-axis means that if a point (x, y) is on the graph, then (-x, y) is also on the graph, typical of even functions.
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Increasing function
An increasing function on a graph rises from left to right, meaning as x increases, y increases, which can be identified by the slope being positive in that interval.
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Decreasing function
A decreasing function on a graph falls from left to right, meaning as x increases, y decreases, indicated by a negative slope in that interval.
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Maximum point
A maximum point on a graph is the highest y-value in a given interval, such as the vertex of an upward-opening parabola.
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Minimum point
A minimum point on a graph is the lowest y-value in a given interval, like the vertex of a downward-opening parabola.
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Area of a triangle with coordinates
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula (1/2)| (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) |.
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Perimeter of a polygon in coordinate plane
The perimeter of a polygon in the coordinate plane is the sum of the distances between consecutive vertices, calculated using the distance formula for each side.
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Equation of a line perpendicular to another
To find the equation of a line perpendicular to another, use the negative reciprocal of the original slope and pass through a given point, then apply point-slope form.
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Equation of a line parallel to another
To find the equation of a line parallel to another, use the same slope as the original line and pass through a given point, then use point-slope form.
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Completing the square for vertex form
Completing the square rewrites a quadratic equation from standard form to vertex form by adding and subtracting the square of half the coefficient of x inside the equation.
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Discriminant for quadratic graphs
The discriminant is b^2 - 4ac in a quadratic equation ax^2 + bx + c = 0, indicating the nature of the roots: positive for two real roots, zero for one real root, and negative for no real roots, affecting the graph's x-intercepts.