Quadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.

Transcription

1 Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go over the parent unction or a variet o algebraic unctions in this section. It is much easier to see the eects dierent constants have on a particular unction i we use the parent unction. We will begin with quadratics. Observe the ollowing regarding a quadratic unction in standard orm. = a( h) + k = Notice that in the equation above, the h and k values are zero, while the value o a is one. This gives ou the parent unction or all quadratics. Everthing else is merel a manipulation o the parent unction. = ( 3) = ( + 3) The graph o the unction shits right 3. The graph o the unction shits let 3. The number inside the parenthesis makes the graph shit to the let or right. Remember P.L.N.R., Positive Let Negative Right, tells about the horizontal shit needed to graph the unction. I the unction above is ( ), the unctions below would be ( 3) and ( + 3) respectivel. This is important to know, because in the uture, ou will be required to graph unctions based solel on the picture provided. No equation will be given. You must rel solel on our knowledge o translating graphs.

2 = Once again, the parent unction is illustrated above, and translations o it below. = + = The graph o the unction shits up. The graph o the unction shits down. In these eamples, the k value is what is changing. The value o k dictates a vertical shit o the unction. In this case, consider the parent unction as being. Given no inormation regarding the speciic equation o the unction, the equations or these two translations o = ( ) ( ) are ( ) +, and ( ). Now, on the let we have the opposite o the parent unction. In this particular eample, the value o a, in the standard orm is -1. A negative relects the graph o the unction about the horizontal ais. Once again, i the parent unction given is reerred to as ( ), this unction is ( ).

3 We have seen how to graph a unction b shiting the parent unction. You ma have noticed that we graphed ( ), but not ( ). The reason we did not see ( ), is because this is the graph o an even unction. That means that i a were plugged in to the unction, it would make no dierence. The outcome would be the same. However, i we are dealing with a dierent tpe o unction, one that was not even, ( ) would cause the graph o the unction to relect about a vertical ais. In other words, i ( ) makes a graph lip upside down, ( ) would make the graph lip rom right to let, or let to right, whatever ma be the case. Lets see how diering values o a, h and k will cause various shits o the unction. = a( h) + k Once again, take note o the parent unction = = ( + ) + 3 ( ) = 3 + This graph opens down, and shits let, up 3. This graph opens up and shits right 3, and up I this graph is a translation o the unction It would b written as ( ) ( ) I this graph is a translation o the unction It would b written as +. ( ) 3 ( ) We will be graphing unctions using onl ( ) in the translations o unctions section.

4 = a( h) + k = Here we will see how the value o a or the quadratic unction in standard orm aects the graph o the unction. To illustrate this, we will look at the graph o a parabola that has its verte on the origin. = 4 = 1 4 This graph seems ver narrow, but what is actuall happening, is the value o the unction is increasing ver rapidl. The values are increasing at 4 times their normal rate. The rapid increase causes the graph to appear narrow. This graph is wider than the parent unction. In this case, the values o the unction are increasing at ¼ their normal rate, causing a more gradual increase. As ou can see, i the value o the leading coeicient is a whole number, the values o the graph will increase rapidl causing a narrow and steeper curve. In contrast, i the leading coeicient is a raction, the values o the unction will increase mildl, causing a more gradual curve.

5 Describe the movement o each o the ollowing quadratic unctions. Describe how each opens and i there is an horizontal or vertical movement. Be sure to state how man spaces it moves, or eample: This graph opens down, and shits let, up A) = 3( 4) + B) = ( + 3) 8 C) = ( ) D) 1 = + 3 E) = ( + 5) + 6 F) ( ) = G) = ( ) + H) = 3( + 6) + 8 I) ( ) = 4 3 J) = 3 K) ( ) = L) 5 = + 8 As ou describe the graphs o the quadratic unctions above, ou wrote that it shits to the let or right, and up or down. What is actuall shiting?

6 Write the equation or a quadratic unction in = a ( h) + k orm that opens down, shits let 3 and up 7. Write the equation or a quadratic unction in = a ( h) + k orm that opens up, shits right 4 and down. Write the equation or a quadratic unction in = a ( h) + k orm that opens up, and onl shits down 4. Write the equation or a quadratic unction in = a ( h) + k orm that opens down and shits to the let 8 spaces. Write the equation or a quadratic unction in = a ( h) + k orm that opens down and shits up 7. Is a quadratic unction a one-to-one unction? Wh or wh not? What does this tell ou about the inverse o a quadratic unction?

7 Match the appropriate graph with its equation below. Eplain wh each o our solutions is true. A B C D E F = 1 + ) ( ) 1) ( ) ( ) = 3) ( ) ( ) = ) ( ) = ( ) + 1 5) ( ) = 3( + ) 6) ( ) ( ) = 3 +

8 Graph each o the ollowing unctions. You ma need to use an ais o smmetr to graph some o these. Label the verte, -intercept, and all -intercepts. A) ( ) = ( 3) + 1 B) ( ) ( ) = C) ( ) = ( ) D) ( ) ( ) =

9 E) ( ) = + 3 F) ( ) ( ) = G) ( ) = ( 4) H) ( ) ( ) = 1 8

10 The quadratic unction given b the equation ( ) ( ) o. = has an ais o smmetr The quadratic unction given b the equation ( ) ( ) o. = has an ais o smmetr The quadratic unction given b the equation ( ) ( ) o. = a h + k has an ais o smmetr Considering our answers to the previous questions, we can conclude that the ais o smmetr or an quadratic unction is given b the value o the. Wh does the ais o smmetr look as though we are saing equals a number ( = # )? Wh is it sometimes necessar to graph a quadratic unction using the ais o smmetr?