# Determine the real numbers a, c if the points (1;2) and (0;3) are on the graph of f(x)=ax^2+x+c.

justaguide | College Teacher | (Level 2) Distinguished Educator

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We are given that the points (1,2) and (0, 3) are on the graph of f(x)=ax^2+x+c.

We can write the following equations:

3 = a*0 + 0 + c = c

and  2 = a +1 +c = a + c + 1

Now from the first equation we have c = 3.

Substitute this in 2 = a +1 +c

=> 2 = a + 1 + 3

=> a = -2

Therefore the real numbers a and c are a = -2 and c = 3.

neela | High School Teacher | (Level 3) Valedictorian

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To determine the real numbers a, c if the points (1;2) and (0;3) are on the graph of f(x)=ax^2+x+c.

(1,2) is on ax^2+x+c.

=> f(1) = a*1^2+1+c = 2.

Or a+1+c = 2

a+c = 2-1= 1.

a+c = 1......(1)

(0,3) is on ax^2+x+c.

=> f(0) = a*0^2+0+c = 3.

=> c = 3....(2)

We put c = 3 in (1): a+c = 1. Or a+3 = 1. So a = 1-3 = -2.

Therefore a = -2, c = 3.

So ax^2+x+c = -2x^2+x+3.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

If the graph of the function f(x) is passing through the given points, then the coordinates of these points have to verify the expression of the function.

The point A(1 , 2) is on the graph if and only if:

f(1) = 2

We'll substitute x by 1:

f(1) = a + 1 + c

a + 1 + c = 2

The point B(0 , 3) is on the graph if and only if:

f(0) = 3

We'll substitute x by 0:

f(0) = c

c = 3

We'll substitute c = 3 in:

a + 1 + c = 2

a + 1 + 3 = 2

We'll subtract 4 both sides:

a = 2 - 4

a = -2

The expression of f(x) is: f(x) = -2x^2 + x + 3