The answer is a = 1, b = 2, and c = -3. How just by look at the points? C is intuitive, but I don't get the other points.

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3 Answers
Nov 14, 2017

if a>0 =>"smile" or uuu like => min
if a<0 =>"sad" or nnn like => max

x_min=(-b)/(2a)
y_min=y_((x_min))

x_(1,2)=(-b+-sqrt(b^2-4ac))/(2a)

Explanation:

just to explain x=(-b)/(2a):

if you want to find the x_min or x_max you do y'=0, right?

Now, because we are dealing with the form of
y=ax^2+bx+c
the differentiate is always in the form of
y'=2ax+b

now we say (in general):
y'=0
=> 2ax+b=0
=> 2ax=-b
=> x=(-b)/(2a)

So as we see, the x_max or x_min is always x=(-b)/(2a)

Nov 14, 2017

a=1,b=2,c=-3

Explanation:

"one possible approach"

c=-3larrcolor(red)"y-intercept"

• " sum of roots "=-b/a

• " product of roots "=ca

"here the roots are "x=-3" and "x=1

"that is where the graph crosses the x-axis"

rArr-3xx1=carArrca=-3rArra=-3/(-3)=1

rArr-b/a=-3+1=-2rArrb=2

rArry=x^2+2x-3
graph{x^2+2x-3 [-10, 10, -5, 5]}

Nov 14, 2017

Bit wordy but work you way through it. Full explanation given.

Explanation:

Given the standardised form y=ax^2+bx+c

The curve at the bottom has the special name (what doesn't in maths) of Vertex.

If there are x-intercepts (where the graph crosses the x-axis) then the Vertex value of x is 1/2 way between

Looking at the graph the x-intercepts are at x=-3 and x=1

So the x value of the vertex is the average

x_("vertex") = (-3+1)/2=-1

This is what relates x_("vertex") to the equation.

Write as y=a(x^2+b/ax)+c" "......................Equation(1)

x_("vertex")=(-1/2)xxb/a

-1=(-1/2)xxb/a

Divide both side by (-1/2)

color(brown)(2=b/a)

Substitute into Equation(1) giving

y=a(x^2+2x)+c" "....................Equation(1_a)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Lets pick a known point.
I choose the left hand x-intercept ->(x,y)=(-3,0)

Known that c=-3

Substitution into Equation(1_a)

y=a[color(white)("dd")x^2color(white)("dd")+color(white)("d")2xcolor(white)(()^2)]+c

0=a[(-3)^2+2(-3)]-3

Add 3 to both sides and simplify the brackets

3=9a-6a

color(brown)(3=3a=>a=1)

Thus color(brown)(2=b/a->2=b/1=>b=2)

y=ax^2+bx+c

color(magenta)(y=x^2+2x-3)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Note that:

y=a(x^2+b/ax)+c" ".........Equation(1)

is the beginnings of completing the square.