What is the quadratic function of a graph if the points are (-2, -4) (-1, -2.5) (0, 0) (1, 3.5) (2, 8)?

Answer 1

It can be guessed since we deal with easy numbers, but let's approach this problem theoretically.

The general equation of a quadratic function looks like this: #y=ax^2+bx+c# There are three unknown coefficients here - #a#, #b# and #c#

Let's use three points out of 5 given to determine these three coefficients.

Point #(-2,-4)# results in equation #-4=a(-2)^2+b(-2)+c# or, simplifying this, (1) #-4=4a-2b+c#
Point #(-1,-2.5)# results in equation #-2.5=a(-1)^2+b(-1)+c# or, simplifying this, (2) #-2.5=a-b+c#
Point #(0,0)# results in equation #0=a(0)^2+b(0)+c# or, simplifying this, (3) #0=c#

Equations (1), (2) and (3) constitute a system of 3 linear equations with three unknown variables. Let's solve it by substitution.

Step 1. Substitute #c=0# from equation (3) into equations (1) and (2): (1) #-4=4a-2b# (2) #-2.5=a-b#
Step2. Simplify the equation (1) by dividing left and right sides by 2: (1) #-2=2a-b# Solve it for #b#: #b=2a+2#
Step 3. Substitute an expression for #b# into equation (2): (2) #-2.5=a-2a-2# or #a=0.5#
Step 4. Find the value of #b#: #b=2*0.5+2=3#
Together with previously determined #c=0#, we have an equation: #y=0.5x^2+3x#

All we have to do is to check that two other points specified in the problem lie on this graph.

Point #(1, 3.5)#: #0.5*1^2+3*1=0.5+3=3.5# (check!) Point #(2,8)#: #0.5*2^2+3*2=0.5*4+6=2+6=8# (check!)
So, the answer to this problem is #y=0.5x^2+3x#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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