1. `y=x^2+3x-4`, find Directrix of a function
Solution:
`y=x^2+3x-4`
1. Vertex :
`y=x^2+3x-4`
Method-1: Find vertex using polynomial form
Comparing the equation x^2+3x-4 with `ax^2+bx+c`, we get
`a=1,b=3,c=-4`
`h=(-b)/(2a)=(-3)/(2 * 1)=-3/2`
Now, substitute value of h in f(x), to find value of k
`k=f(h)=f(-3/2)=(-3/2)^2+3(-3/2)-4`
`:. k=1/2-9/2-4`
`:. k=-25/4`
Vertex `=(h,k)=(-3/2,-25/4)`
Method-2: Find vertex using vertex form `y=a(x-h)^2+k`
Completing the Square
`x^2+3x-4`
`=1 (x^2+3x-4)`
The coefficient of the x is `3`, so now we divide this by 2 : `(3 -: 2 = 3/2)`
and square it `(3/2)^2=9/4`. So we add and subtract `9/4`
`=1 (x^2+3x + 9/4 - 9/4 - 4)`
`=1 [(x^2+3x+9/4) -25/4]`
`=1[( x + 3/2 )^2 -25/4 ]`
`:. y=1(x-(-3/2))^2+(-25/4)`
Now compare with `y=a(x-h)^2+k`, we get
`a=1,h=-3/2,k=-25/4`
Vertex `=(h,k)=(-3/2,-25/4)`
If `a<0` then the vertex is a maximum value
If `a>0` then the vertex is a minimum value
Here `a=1>0`
So minimum Vertex = `(h,k)=(-3/2,-25/4)`
2. Focus :
Find `p`, distance from the vertex to a focus of the parabola
`p=1/(4a)=1/(4*1)=1/4`
Focus `=(h,k+p)=(-3/2,-25/4+1/4)=(-3/2,-6)`
3. Directrix :
Directrix `y=k-p=-25/4-1/4=-13/2`
4. Graph :
some extra points to plot the graph
`y=f(x)=x^2+3x-4`
`f(-5)=(-5)^2+3(-5)-4=25-15-4=6`
`f(-4)=(-4)^2+3(-4)-4=16-12-4=0`
`f(-3)=(-3)^2+3(-3)-4=9-9-4=-4`
`f(-2)=(-2)^2+3(-2)-4=4-6-4=-6`
`f(-1)=(-1)^2+3(-1)-4=1-3-4=-6`
`f(0)=(0)^2+3(0)-4=0-4=-4`
`f(1)=(1)^2+3(1)-4=1+3-4=0`
`f(2)=(2)^2+3(2)-4=4+6-4=6`
`f(3)=(3)^2+3(3)-4=9+9-4=14`
graph
This material is intended as a summary. Use your textbook for detail explanation.
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