y=4 x 2 +12x+9

Question

a) If graph opens up or down
b) Vertex
c) Axis of symmetry
d) x and y intercepts
e) Sketch a graph

Answer 1

y=4x² +12x + 9

 

1.) the Graph opens up because the numerical coefficient of the quadratic term ( the 4) is positive.

2.) You may have been taught the vertex formula at some point, if not spoiler alert! Use 

 

x=-b/2a to find the x value of the vertex, then substitute what you get back into the original function to find the y value.

Remember that  y=ax²+bx+c, for your problem, b=12, so -b = -12, and a=4.

x=-b/(2a) becomes,

x = -12/(2*4) 

x = -12/8 = -3/2

 

Now substitute:

y=4x² +12x + 9

y=4(-3/2)² +12(-3/2) + 9

y=4(9/4) +6(-3) + 9

y= 9 -18 +9

y=0

so the vertex is at (-3/4, 0)

 

3.) The axis of symmetry goes through the vertex and is a vertical line.  It's equation is just x=-3/2.

 

4.) for your x and y intercepts half of the work is done! We already know that the graph touches the x-intercept when y=0, and that we determined happens at  x=-3/4!  so the x-intercept is at the point (-3/4, 0)

 

To find the y-intercept, we just need to know what y is when x=0!

 Given:  y=4x² +12x + 9,  Let x=0, 

y=4(0)² +12(0) + 9

y= 0+0+9

y=9, 

so the y-intercept is the point (0,9)

 

5.) Now that you have the key points, graphing wont be that bad.  Put a point at the points that you have.  Sketch the axis of symmetry with a dotted line.  Now determine some easy points and remember that for every point on one side of the axis of symmetry, there is another one the same distance from it, but on the other side.  So if you even measure with your fingers how far (0,9) is from your axis of symmetry, you can make another point the same distance from the axis of symmetry, but on the other side of it.  Also, try another easy point, maybe let x=-3, then 

y=4(-3)² +12(-3) + 9

y=4(9)+-36 + 9

y = 36 -36 + 9

y=9, not so bad, so you have the point (-3,9). This is a pretty stretch out curve, so you might not get more points to fit on your paper.  Go ahead and sketch a nice curve that goes through the points that you do have.  Don't forget to put arrows on the curves to show that you know that they keep on going up forever.

 

Good Luck!

 

Miss Linda B