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Test PrepACTMathGraphs and Coordinate Geometry

ACT Math Skill Review: Graphs and Coordinate Geometry

You will be expected to answer questions involving both linear and quadratic equations as well as their graphs. Therefore, you will need to understand the basics of the coordinate plane.

The Cartesian Grid

The Cartesian Grid can be broken up into four quadrants with respect to the x-axis and y-axis. The signs of x and y change depending on which quadrant the point lies. The graph below labels each of the four quadrants. Every real point (x, y) has a place on this grid. For instance, the point A (3, 1) can be found by counting over on the x-axis three places to the right of the origin (0, 0) and then counting on the y-axis, one place up from the origin.

graph1.jpg

Always remember: moving to the right on a grid means x is becoming more positive, while moving up on a grid means y is becoming more positive.

Finding the Midpoint between Two Points

You will be expected to know how to find the midpoint of line segments in the coordinate plane. The midpointeq1.PNGis the average of the Xs and the average of the Ys. We can write this out as:

eq2.PNG andeq3.PNG

For instance, if AB has endpoints A (4, 5) and B (2, 3), then we can determine the coordinates of the midpoint as follows:

eq4.PNG

Finding the Distance between Two Points

The Pythagorean Theorem can be used to find the distance between any two points in the coordinate plane. Take the figure below.

graph2.jpg

Points A (1, 1), B (4, 4) and C (4, 1) form a right triangle, ΔABC, where AC = 3 and CB = 3. Thus, ΔABC is an isosceles right triangle. Applying the Pythagorean Theorem, we would get:

eq5.PNG

Therefore, the distance between two points,eq6-(1).PNG,in the coordinate plane can be summarized by the formula:

eq7-(1).PNG

Equation of a Line

The equation of a line can be found by using the following formula:

eq8.PNG

In this formula, x and y are represented by the point (x, y), m is the slope of the line (how sharply a line is inclining or declining), and b is the y-intercept (the point where a line crosses the y-axis).

Slope

The slope of a line can be found if you know two points that sit on the line. It is defined as:

eq9.PNG

For example, if you are given the equation y = 2x + 1, you will know that the slope is 2 and the y-intercept is 1. That means that the line crosses the y-axis at a point 1 above the origin (y = 1 and x = 0). An easy way to conceptualize slope is as the fraction 21, where 2 represents the direction of movement on the y-axis and 1 represents the direction of movement on the x-axis. Thus, for every 2 points you move up along the y-axis, you must move 1 point along the x-axis.

Key Relationships

  • A positive slope represents an inclining line (y values increase from left to right).
  • A negative slope represents a declining line (y values decrease from left to right).
  • Two lines are parallel if they have equal slopes. For example, the lines y = 3x + 1 and y = 3x – 2 are parallel because both lines have a slope of 3.
  • Two lines are perpendicular when the product of their slopes equals -1. For example, the lines y = 2x + 1 and y = (-12)x – 3 are perpendicular because 2 times -12 = -1.
  • A horizontal line has a slope of 0. Where the line crosses the y-axis will determine its placement. Such a line would be represented by an equation such as y = 3.
  • A vertical line has an undefined slope. Where the line crosses the x-axis will determine its placement. For example, a line represented by the equation, x = 2, would be vertical.

Graphs of Quadratic Functions

You may need to be able to identify some of the features on the graph of a quadratic equation, such as its highest or lowest point, its solutions and its direction. The equation of a quadratic function is expressed as:

eq12.PNG

The graph of a quadratic function is called a parabola. A parabola is a U-shaped curve that can open upward or downward depending on the sign of a. If a > 0, then the graph will open upward. If a < 0, then the graph will open downward.

graph3.jpggraph4.jpg

Graphs of Ellipses and Circles

It is a good idea to familiarize yourself with the equations of a circle and an ellipse.

The equation for a circle is

GraphAndCoord1.PNG
where (h, k) is the center of the circle and r is the radius, as shown below.
GraphAndCoord2.PNG
The equation for an ellipse is
GraphAndCoord3.PNG
where (h, k) isi the center fo the ellipse, 2a is the horizontal axis (width) and 2b is the vertical axis (height), as shown below.
GraphAndCoord4.PNG

Transformations

Translation

A translation is described as a linear movement that does not involve any rotations or reflections. In the figure below, the line segment has been translated to two units right in the positive x-direction.

g1.PNG

Rotation

When a figure is rotated, it is turned around a central point, or point of rotation. The first rectangle below has been rotated 90° to create the second rectangle.

g2.PNG

Reflection

When a figure is reflected, its mirror image is produced with respect to a line. The triangle on the left below has been reflected about line l to create the triangle on the right. The two triangles are mirror images of each other.

g3.PNG

Symmetry

When a figure can be folded such that each half matches the other exactly, the figure is said to possess a degree of symmetry. The line on which the figure is folded in order to get the equal halves is called a line or axis of symmetry. Line l is one axis of symmetry in the figure below.

Graph3.PNG

When a figure is rotated and the resulting figure is the same as the original figure, the figure is said to possess symmetry about a point (point of symmetry). A rotation of 180° of the figure below will yield the same figure.

Graph4.PNG

Note: Symmetry about a point and symmetry about a line are different properties. A given figure may have either type of symmetry, both types of symmetry or neither type.

Examples


Answers and Explanations

  1. The correct answer is E. The equation for a line is y = mx + b, where m is the slope and b is the y-intercept. Point B is the y-intercept, so we know that y = 2 at the point where the line crosses the y-axis. All of the answer choices have b as equal to 2, so that information doesn’t help us eliminate any answer choices. If we did a quick drawing, we’d see that the line points down to the right, so the slope must be negative. That allows us to eliminate A, C and D. At this point, we could either plug both sets of coordinates into the two remaining answer choices and see what works or we could figure out the slope. The slope is equal to the change in y (the rise) over the change in x (the run). So,ans11.PNG=-1. Or,Capture.PNG-1. The order doesn’t matter, as long as you are consistent. In choice B, the slope iseq.PNGwhile in choice E, the slope is -1.
     
  2. The correct answer is G. Like many coordinate geometry questions, this one involves triangles. A quick drawing can crack this question wide open.

    graph.PNG

    It’s straightforward to identify the length of the base (b) as 3, but the height takes a little more thought as it must be perpendicular to the base. Dropping a height down from point (2, 4) to point (2,-2) does the trick, however, making it easier to see that the height (h) is equal to 6. Plugging these two values into the formula for the area of a triangle and you get:eql.PNG


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