Mathematics Study Guide for the ACT
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Graphing
Graphing helps us visualize and analyze data, functions, and relationships between variables. We can plot points, lines, and curves on a coordinate grid, a two-dimensional surface that consists of two perpendicular lines, called axes, labeled as \(x\) and \(y\). The horizontal line is the \(x\)-axis, and the vertical line is the \(y\)-axis. The intersection point of both axes is called the origin.
The labels on the axes represent the values of the variables being plotted. The \(x\)-axis is labeled with numbers increasing from left to right, while the \(y\)-axis is labeled with numbers increasing from bottom to top.
Points
A point is a fundamental concept of geometry and coordinate geometry. It is simply a location in space. We name a point using any capital letter, such as \(A\), \(B\), or \(C\). Although it is represented by a dot on a diagram or graph, we imagine a point to have no size or shape. It is used to define lines, planes, and angles.
Locating Points
To locate a point on the coordinate grid, we use ordered pairs in the form \((x,y)\). The first number, \(x\), represents the horizontal distance the point is from the origin, while the second number, \(y\), represents the vertical distance the point is from the origin.
We can plot the point by starting at the origin, \((0,0)\), and moving horizontally \(x\) units and then, from there, moving vertically \(y\) units. To plot the point \((2,3)\) on the coordinate plane, we start at the origin and move two units right (along the \(x\)-axis) and three units up (parallel to the \(y\)-axis). The coordinates (ordered pair) of the point is \((2,3)\).
Remember that a point with negative \(x\) coordinates moves to the left and negative \(y\) coordinates moves down.
Distance Between Two Points
The distance between two points on the coordinate grid is the square root of the sum of the squares of the differences between the \(x\) and \(y\) coordinates of the two points. The distance formula is:
\[D = \sqrt{(y_2 - y_1)^{2} + (x_2 - x_1)^{2}}\]where \(D\) is the distance, and \((x_1,y_1)\) and \((x_2,y_2)\) are the coordinates of the two points.
For example, the distance between the points \((-1,3)\) and \((2,7)\) is five units. Here’s how we know that:
\[D = \sqrt{(y_2 - y_1)^{2} + (x_2 - x_1)^{2}}\] \[D = \sqrt{(7-3)^{2} + (2-(-1))^{2}}\] \[D = \sqrt{4^{2} + 3^{2}}\] \[D = \sqrt{16 + 9}\] \[D = \sqrt{25}\] \[D = 5\]You will also need to be familiar with reflecting, rotating, and translating points for the ACT.
Lines
A line is a collection of points that extends infinitely in both directions. It can be represented visually as a graph in the coordinate grid or as a mathematical equation. A line is an equation with two variables, usually \(x\) and \(y\).
Slope
The slope of a line is the measure of its steepness. It also can be thought of as how much a line rises or falls over a given horizontal distance. The slope of a line is defined as the ratio of the change in \(y\) coordinates to the change in \(x\) coordinates between any two points on the line. The change in \(y\) coordinates is known as rise and the change in \(x\) coordinates is known as run. We denote the slope of a line with the letter \(m\), and it can be calculated using this formula:
\[m = \frac{y_2-y_1}{x_2-x_1}\]Generally, the equation of a line is given in one of the three forms discussed below.
Slope-Intercept Form
This form follows the equation:
\[y = mx + b\]where \(m\) is the slope of the line and \(b\) is the \(y\)-intercept (the point where the line cuts through the \(y\)-axis). This form is convenient for graphing, as it allows you to determine the slope and \(y\)-intercept of a line directly from its equation.
Point-Slope Form
The point-slope form is:
\[y - y_1 = m(x-x_1)\]where \(m\) is the slope and \((x_1,y_1)\) is a point on the line. This form is useful when you know a point on the line and its slope, and you need to find the equation of the line.
Standard Form
The standard form is:
\[Ax + By = C\]where \(A\), \(B\), and \(C\) are constants. This form is useful for certain types of problems, such as finding the distance between two parallel lines or the intersection of two lines.
Parallel and Perpendicular Lines
Two lines are parallel if they never intersect. They have the same slope. Two lines are perpendicular if they intersect at a right angle (\(90^{\circ}\)). Their slopes are negative reciprocals of each other.
Using the slopes of lines, you can determine whether two lines are parallel or perpendicular. For example, let line \(L_1\) have equation \(y = -5x + 6\) and line \(L_2\) have equation \(y = -5x -3\). As you can see, both lines are written in the slope-intercept form, and both have the same slope, \(-5\). Thus, line \(L_1\) is parallel to line \(L_2\).
Using Graphs
Using graphs, we can show visual representation of data and information. They allow us to analyze information, draw conclusions, and make predictions based on patterns and trends we observe on the graph.
Analyzing and Drawing Conclusions
Using linear and quadratic graphs, we can make predictions about what might happen in the future. For example, if we are given the following graph of temperature over the course of seven days in a certain month, we can use the graph to predict the temperature on the eighth day.
If you look at the trend of the graph, you will see that each day the temperature increases by \(0.5^{\circ}\text{C}\). On the seventh day, the temperature was \(24^{\circ}\text{C}\). If the trend continues, the temperature on the eighth day will be \(24 + 0.5 = 24.5^{\circ}\text{C}\).
Additionally, graphs can help us compare data and identify relationships between different variables.
Identifying Characteristics of Graphs
As mentioned above, the slope-intercept form of a line is given by the equation \(y = mx + c\). The value of \(m\) represents the slope of the graph. We can see the change in the \(y\) variable with respect to the change in the \(x\) variable. If the value of \(m\) is positive, the graph of the line is sloping upward and if the value of \(m\) is negative, the line is sloping downward. The value of \(c\) is the \(y\)-intercept. It shows the value of \(y\) at the initial point (when \(x=0\)).
Moreover, graphs can also help us identify features such as the \(\boldsymbol{x}\)-intercept (\(x\)-axis cutting point), \(\boldsymbol{y}\)-intercept (\(y\)-axis cutting point), and asymptotes (straight lines or curves that a graph approaches but never touches or crosses).
You can also work with the graphs of parabolas (quadratic functions) and find things like the vertex and axis of symmetry.
Matching Equations and Functions
Matching equations and functions requires the ability to identify a function whose graph is a translation of another function by specific amounts horizontally or vertically. For example, consider the functions \(f(x)=x^2\) and \(g(x) = (x+2)^{2} + 3\). The function \(g(x)\) is the function \(f(x)\) translated two units left and three units up:
Relating Graphs to Situations
To relate graphs to real-life situations, you must be able to understand how graphs represent data and how changes in data are reflected in a graph.
If you are given the graph of a car’s speed over time, you can analyze the graph as the car accelerates and eventually comes to a stop. As the car begins to accelerate, the speed increases, and the graph will show a linear positive slope. As the car reaches its top speed, the slope of the linear graph will level off. It shows a constant speed. Finally, as the car begins to brake, the speed decreases, and the graph will show a negative linear slope.
Note: The graph of the trajectory of a ball thrown into the air resembles the graph of a quadratic function (a parabola).
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