1. First we are going to find the vertex of the quadratic function
. To do it, we are going to use the vertex formula. For a quadratic function of the formÂ
, its vertexÂ
is given by the formulaÂ
;Â
.
We can infer from our problem thatÂ
andÂ
, sol lets replace the values in our formula:
Now, to findÂ
, we are going to evaluate the function at
. In other words, we are going to replaceÂ
with -2 in the function:
So, our first point, the vertex
of the parabola, is the pointÂ
.
To find our second point, we are going to find the y-intercept of the parabola. To do it we are going to evaluate the function at zero; in other words, we are going to replaceÂ
with 0:
So, our second point, the y-intercept of the parabola, is the point (0,1)
We can conclude that using the vertex (-2,-7) and a second point we can graphÂ
as shown in picture 1.
2. The vertex form of a quadratic function is given by the formula:Â
where
is the vertex of the parabola.
We know from our previous point how to find the vertex of a parabola.Â
andÂ
, so lets find the vertex of the parabola
.
Now we can use our formula to convert the quadratic function to vertex form:
We can conclude that the vertex form of the quadratic function is
.
3. Remember that the x-intercepts of a quadratic function are the zeros of the function. To find the zeros of a quadratic function, we just need to set the function equal to zero (replaceÂ
with zero) and solve forÂ
.
To solve forÂ
, we need to factor our quadratic first. To do it, we are going to find two numbers that not only add up to be equal 4 but also multiply to be equal -60; those numbers are -6 and 10.
Now, to find the zeros, we just need to set each factor equal to zero and solve for
.
andÂ
andÂ
We can conclude that the x-intercepts of the quadratic function
are the points (0,6) and (0,-10).
4. To solve this, we are going to use function transformations and/or a graphic utility.
Function transformations.
- Translations:
We can move the graph of the function up or down by adding a constantÂ
to the y-value. IfÂ
, the graph moves up; ifÂ
, the graph moves down.
- We can move the graph of the function left or right by adding a constantÂ
to the x-value. IfÂ
, the graph moves left; ifÂ
, the graph moves right.
- Stretch and compression:
We can stretch or compress in the y-direction by multiplying the function by a constantÂ
. IfÂ
, we compress the graph of the function in the y-direction; ifÂ
, we stretch the graph of the function in the y-direction.
We can stretch or compress in the x-direction by multiplyingÂ
by a constantÂ
. IfÂ
, we compress the graph of the function in the x-direction; ifÂ
, we stretch the graph of the function in the x-direction.
a. TheÂ
value ofÂ
is 2; theÂ
value ofÂ
is -3. SinceÂ
is added to the whole function (y-value), we have an up/down translation. To find the translation we are going to ask ourselves how much should we subtract to 2 to get -3?
Since
, we can conclude that the correct answer is: It is translated down 5 units.
b. Using a graphing utility to plot both functions (picture 2), we realize thatÂ
is 1 unit to the left ofÂ
We can conclude that the correct answer is: It is translated left 1 unit.
c. Here we have thatÂ
isÂ
multiplied by the constant term 2. Remember that We can stretch or compress in the y-direction (vertically) by multiplying the function by a constantÂ
.
Since
, we can conclude that the correct answer is: It is stretched vertically by a factor of 2.