Showing posts with label quadratic. Show all posts
Showing posts with label quadratic. Show all posts

Quadratic Inequalities

Quadratic inequalities can be solved in a number of ways. We will focus on solving them by graphing first and then by using sign charts.  It is important to understand what solutions to these inequalities look like before learning the quick and easy method for solving them.

Method 1: Solve by Graphing

   Step 1: Place the inequality in standard form with zero on one side.
   Step 2: Graph the quadratic equation.
   Step 3: Shade the x-values that produce the desired results.
   Step 4: Convert the shading to interval notation.

Solve and express the solution set in interval notation.
In the above example we shaded the x-values for which the graph was above the x-axis. If the problem asked us to solve
or for what x-values is the quadratic less than or equal to 0, then the solution would have been all x-values in the interval [1, 3] - the x-values where the graph is below the x-axis. Notice that the numbers on the x-axis x = 1 and x = 3 separate the positive and negative y-values on the graph, these happen to be the x-intercepts.  The x-intercepts or the zeros of a polynomial are called critical numbers.

Solve by graphing.
Remember to use open dots for strict inequalities < or > and closed dots for inclusive inequalities.

Often it is quite tedious and difficult to graph each inequality when trying to solve them.  There is a shorter method and it involves sign charts.  The idea is to find the critical numbers, the x-values where the y-values could change from positive to negative and create a sign chart to determine which intervals to shade on the x-axis.

Method 2: Solve using sign charts.

   Step 1: Place the inequality in standard form with zero on one side.
   Step 2: Find the critical numbers (for quadratics - the x-intercepts.)
   Step 3: Create a sign chart by determining the sign in each interval bounded by the critical numbers.
   Step 4: Use the sign chart to answer the question.

Solve and graph the solution set.
Find the critical numbers by setting the quadratic expression equal to zero and solve.
 Determine the results + or - in each interval bounded by the critical numbers by testing values in each interval.
Use the sign chart to answer the question. In this case we are looking for the x-values that produce negative results as indicated by the inequality < 0 "less than zero" in the original question.

When testing values in the intervals created by the critical numbers the actual value is not necessary, we are only concerned with its sign.  The sign + or - will be the same for any value in the interval so you may choose any number within the interval when testing.
Solve and graph the solution set.
Tip: Save some time and just determine if the corresponding y-value is positive or negative. If the polynomial factors then use the factors to determine if the interval will produce positive or negative y-values. There is no need to find the actual values.
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Graphing Parabolas

At this point in our study we should be able to find x- and y-intercepts and solve any quadratic equation. Now we will learn an easy method used to graph them.

The graph of a quadratic equation is called a parabola.
One of our basic functions
can be graphed by plotting points. We do this by choosing about five x-values and finding their corresponding y-values.


The more points we plot the easier it is to see that the graph is u-shaped. The vertex, in this case, is the point where the graph changes from decreasing to increasing, or the point with the smallest y-value. Here the vertex is (0, 0) which is also the x- and y-intercept. The line x = 0, the y-axis, is the line of symmetry. This is the line where we could fold our paper to see that the two sides of the graph coincide.
Given the graph find the x- and y-intercepts, vertex, a 5th point on the graph and the line of symmetry.

Line of symmetry: x = 1
x-intercepts: (-2,0) and (4,0)
y-intercept: (0, -8)
Vertex: (1, -9)
5th point: (2, -8)

Recall that two points determine a line - this is not the case for parabolas. Parabolas require a minimum of 3 points but we usually want to find at least five points to make a nice graph. Find the vertex, x- and y- intercepts as well as the line of symmetry.
   Step 1: Find the y-intercept, (0, c).
   Step 2: Find the x-intercepts by setting y = 0 and solving for x.
   Step 3: Find the vertex. You can find the x-value of the vertex using the vertex x = -b/(2a).
   Step 4: Graph the points and identify the axis of symmetry.
The domain and range of the above function can be determined by the graph. In the previous problem the domain consists of all real numbers and the range consists of all real numbers greater than or equal to −1. Also, it is useful to note that we have a minimum y-value of −1, this will be an important fact when working the word problems.

Tip: The axis of symmetry of any quadratic function will be the vertical line
When trying to find x-intercepts where the resulting quadratic equation does not factor, simply use the quadratic formula to solve it.

This parabola looks a bit different, notice that it opens down and also notice that the previous parabola opened up. There is an easy test to know which way it opens even before we begin.
Therefore, when asked to graph a parabola, you can get two important pieces of information without doing any work. By inspection you can tell if it opens up or down and you can identify y-intercept.

Graph and label all important points:

Graph and label all important points:

The domain of the previous problem is all real numbers and the range consists of all real numbers greater than or equal to −5. Also, make a note that the minimum y-value is −5. It turns out that not all parabolas have two x-intercepts as we would expect. Sometimes they have only one x-intercept and sometimes none.
Please keep in mind that all quadratic functions have a vertex and a y-intercept. In addition, we will still be able to find another point using symmetry. So in some cases it is acceptable to plot and label only three points.

Graph and label all important points: 

Graph and label all important points: 

Projectile Problem: An object is thrown from a 100 foot building at an initial speed of 44 feet per second. How long will it take to reach the maximum height? What is the maximum height?

Video Examples on YouTube: