Quadratic Functions 2

Transcription

1 Quadratic Functions The general form of a quadratic function is f () = a + b + c or = a + b + c. Basic quadratic: f () = The graph of an quadratic function is called a parabola. In order to graph a basic quadratic function, we will use these -values:, 1, 0, 1, and. The shape we get from this graph is the general shape of all parabolas. All quadratic functions have this overall shape General parabolic shapes: ais of smmetr of the parabola verte of the parabola Interesting Things About Parabolas 1. The -values start to repeat themselves at some point. (This point is called the verte.). The graph is smmetrical. The point at which we can fold our graph in half is the verte. The crease down the middle of the graph is called the ais of smmetr. 3. If we find the verte of the parabola first, the -values on either side of the verte are the same Verte If we put these things together, we can see that if we find the verte of our parabola first, then we can get an idea about what the rest of the data is going to look like

2 because it centers our graph. So, that s how we re going to go about figuring out how to graph parabolas in general. We re going to find the verte first, and then we re going to pick points on each side of that verte. Given f() = a + b + c: 1) The -value of the verte = How to Find the Verte of a Parabola b a (There are 4 pages to this lesson.) Notice, a and b are the numbers onl. Don t include the! This is the front part of the quadratic formula, and since we ve alread talked about the set-up for how the quadratic formula works, it makes sense to do our set-up for the verte formula the same wa: b ( ) = = a ( ) ) The -value of the verte plug what ou get for into a + b + c. Eample 1: Find the verte of the parabola given b f() = a = b = 4 c = 3 Verte = (, ) b ( ) = = = = Now, plug in this value to find! a ( ) = ( ) + 4( ) 3 = ( ) + 3 = + 3 = You can save a lot of time b using our calculator here but ou MUST use the parentheses or else ou ll get a wrong answer ever time ou plug in a negative number. Eample : Find the verte of the parabola given b f() = a = b = c = Verte = (, ) = b a ( = ( ) = ) = 3( ) + 1( ) 5 = + 5 = =

3 Practice Find the verte. 1) f() = ) f() = adfun.htm Verte Form of a Quadratic Equation Find the verte in each of the parabolas below. In the second equation, multipl out the square and simplif first. What do ou notice about the two equations and their vertices? f() = f() = ( + ) 3 = ( + )( + ) 3 Keep going! While the first equation is written in standard form, the second equation is written in a form called verte form. Can ou see wh? Some people get confused b the fact that the -value of the verte is the opposite sign of the number in the parentheses. The question we need to be asking is this: What number do we plug in for to make the parentheses equal to zero? Once ou get, then plug it in to get. sign.) What are the vertices of the following? (The sign isn t the opposite, unlike the f() = ( 3) + 5 f() = 3( +.5) 7.3 Note: Man books refer to = a( h) + k as standard form instead of verte form. Practice Find the verte of the following parabolas. 1) f() = 3( 4) + 6 ) f() = ( + 9) 5 3) f() = 4( + 0.6)

4 How to Graph a Parabola 1. Find the verte. (Remember, finding the verte first helps to center our graph!). For the other -values, pick the previous two integers and net two integers with respect to the -value of the verte. 3. Plug each of those -values into the equation to get the corresponding -values. 4. Plot our points! Eample 1: Graph the parabola given b f() = b ( ) 1) Find the verte. = = = = a ( ) = ( ) + 6( ) + 8 = = ) Pick the -values as integers above and below the -value of the verte. Integers are just the following:, 3,, 1, 0, 1,, 3, In other words, the are what we ve been calling nice numbers. Put verte here in the middle. 3) Plug in those -values to get the corresponding -values for the chart above. f() = Set-up: = ( ) + 6( ) + 8 = = _ = = 4) Plot the points: (There are 4 pages to this lesson.)

5 Graphing Parabolas in Verte Form Remember, when we re graphing a parabola, we want to find the verte first, and then find two other points on either side of the verte to graph so that we get the curved shape we re all familiar with. When a quadratic equation is in verte form, the verte is much easier to find than if the quadratic equation is in standard form. Eample: Graph f() = ( ) + 1 The verte is (, 1). Since the -value for the verte is, then for the other -values, we ll pick 3 and 4 on the left and 0 and 1 on the right. (The two nearest, nice -values to = ). Then, plug in 0, 1, and 3, 4 for and see what we get for f(). Verte! f() = ( ) + 1 = = _ = = In a lot of was, graphing a quadratic equation that is written in verte form is almost easier than graphing one that is written in standard form. Plugging in the other values of (0, 1, and 3, 4) is easier, and this form makes it a little more evident wh parabolas are smmetric. Websites of interest: (Contains an interactive link!)