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Maths-

General

Easy

Question

The vertex of the parabola in the $x y$ -plane above is . Which of the following is true about the parabola with the equation $y equals negative a left parenthesis x minus b right parenthesis squared plus c$ ?

$The vertex is (b, c) and the graph opens upward.$
The vertex is $(b, c)$ and the graph opens downward.
The vertex is $(- b, c)$ and the graph opens upward.
The vertex is $(- b, c)$ and the graph opens downward.

hint Hint:

Hint:

Whenever we change something in $y comma x$ of function $y equals f left parenthesis x right parenthesis$ , then its graph also shifted.

The correct answer is: The vertex is $left parenthesis b comma c right parenthesis$ and the graph opens downward.

Explanation:

We have given the graph of a parabola $y equals a x squared plus c$ with vertex $left parenthesis 0 comma c right parenthesis$

We have to find the vertex and nature of the graph of the equation $y equals negative a left parenthesis x minus b right parenthesis squared plus c$ .

Here we have to use the transformation of the graph. We know that if $f left parenthesis x right parenthesis not stretchy rightwards arrow f left parenthesis x minus a right parenthesis$ , the graph shifted to the right side.

Step 1 of 1:
We have a parabola $y equals a x squared plus c$ with vertex $left parenthesis 0 comma c right parenthesis$
Now the new equation is $y equals negative a left parenthesis x minus b right parenthesis squared plus c$
Here we can see that the function changes as $f left parenthesis x right parenthesis not stretchy rightwards arrow f left parenthesis x minus b right parenthesis$
So, its vertex will shift right side with the unit b
So, The new vertex will be $left parenthesis b comma c right parenthesis$ .
And since the sign of a changes, the parabola opens downward.
Hence, Option(B) is correct.

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