If the parabola represented by f(x) = ax^2 + bx + c passes

@bunnuel i have dited my post for the st I ..it was a typo error i am sorry abt that

it is f(-1) > f(-2)

i am not clear as to how did u get the vertex as x=1

Intersection points of parabola with x-axis are \((-3,0)\) and \((5,0)\):

-----(-3)--0----(5)---, as parabola is symmetric, the x coordinate of the vertex must be halfway between \(x=-3\) and \(x=5\) --> \(x=\frac{-3+5}{2}=1\). As parabola is downward, \(f(x)\) naturally will have it's max values at vertex \(x=1\), \(f(1)\).

As for the typo in the stem. If I is saying: \(f(-1) > f(-2)\), then it's true as \(x=-1\) is closer to \(x=1\), than \(x=-2\), which means that \(f(-1)>f(-2)\).

amazing explanation Bunuel. I would normally have found a, b and c and then solved. Thanks for showing me the way to think more than the way you solved the problem.

HI Bunuel,

Please explain the red part , what i understood from making graph , if we move left from the axis point certainly value of f(x) decreases but if we move right side from the axis where x =1, value of f(x) will increase till certain point and then will decrease, so how can we deduce that value of f(x) is max at x =1

Check below:


Also check parabola chapter HERE.

Hope it helps.

Wonderful approach! thanks a lot!

Hi Bunuel, could you please explain the highlighted part. How do we infer that parabola is downward? According to theory we only know that parabola is downward if a<0.

Hint: put the given three points on the plane,

I got it. We can know it when putting all the 3 points on a plane. I thought there's another special way. Thanks

Concept: Once you find the Line of Symmetry of a Downward Opening Parabola, the X Coordinate at this Vertex will give you the HIGHEST VALUE that f(x) = y can have


Rule: to find the Line of Symmetry of a Parabola (in which everything on the Left Side of the Axis/Line is a "MIRROR REFLECTION" of all the Points on the Right Side), find the MID-POINT between the X-Intercepts that cross the X-Axis


between (-3 , 0) and (+5 , 0) ---- the Line of Symmetry will occur at the Vertical Line of: X = +1

Also, since this is a Downward Opening Parabola, the Vertex at f(+1) will also provide the HIGHEST VALUE of y

I. f(-1) > f(2)


f(-1) is +2 Units to the LEFT of the Highest Value at the Vertex f(+1)

However, f(2) is only +1 Unit to the RIGHT of the Highest Value at the Vertex f(+1)

the Value of f(2) will be GREATER than the Value of f(-1)

NOT True



II. f(1) > f(0)

The Highest Value will occur at the Vertex, when X =+1, because this is a Downward Opening Parabola.

Thus f(1) will be greater than any other Value

TRUE


III. f(2) > f(1)

As stated above, the Highest Value will occur at the Vertex when the X-Coordinate is +1 (at the Line of Symmetry) because this is a Downward Opening Parabola.

thus: f(2) < f(1)

III is NOT True


ONLY II must be True

-B-

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