Find the x and y intercepts, the vertex and the axis of symmetry of the parabola with equation y = - x 2 + 2 x + 3?
Answers
The x-intercepts are the intersection of the parabola with the x-axis which are points on the x-axis and therefore their y coordinates are equal to 0.
0 = - x 2 + 2 x + 3
Factor right side of the equation: -(x - 3)(x + 1)() = 0
x intercepts are: Solve for x: x = 3 and x = -1 ,
The y-intercepts are the intersection of the parabola with the y-axis which is a point on the y-axis and therefore its x coordinates are equal to 0
y intercept is : y = - (0) 2 + 2 (0) + 3 = 3 ,
The vertex is found by writing the equation of the parabola in vertex form y = a(x - h) 2 + k and identifying the coordinates of the vertex h and k.
y = - x 2 + 2 x + 3 = -( x 2 - 2 x - 3) = -( (x - 1) 2 - 1 - 3) = -(x - 1) 2 + 4
Vertex at the point (1 , 4)
You may verify all the above points found using the graph of y = - x 2 + 2 x + 3 shown below.
Here is your answer
For the y intercept, set x to zero in the equation to see that y=3 . Therefore, the y intercept is at (0,3) . Similarly, for the x intercept, set y to zero in the equation and solve for x by factoring the quadratic polynomial:
y=0=x−2x−3=(x−3)(x+1)
So, we see that y is zero when x=−1 or x=3 . Therefore, the x intercepts are (−1,0) and (3,0) .
To find the vertex, we complete the square:
y=−x2+2x+3
y=−(x2−2x+1)+4
y=−(x−1)2+4
(y−4)=−(x−1)2
(x−1)2=−(y−4)
From this form, we see that the parabola has its vertex at (1,4) . Since the axis of symmetry must be parallel to the y axis and passes through the vertex, the axis of symmetry is x=1 . Plotting these results, we get:
Note : image attached
Step-by-step explanation:
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