(a) Does the parabola open upward or downward? upward downward (b) Find the coor

Question

(a)

Does the parabola open upward or downward?

upward

downward

(b)

Find the coordinates of the vertex.

vertex:

,

(c)

Find the x-intercept(s). If there is more than one x-intercept, separate them with commas. If there are no x-intercepts, click on "None".

x-intercept(s):

(d)

Find the equation of the axis of symmetry.

equationofaxisofsymmetry:

(a)

Does the parabola open upward or downward?

upward

downward

(b)

Find the coordinates of the vertex.

vertex:

,

(c)

Find the x-intercept(s). If there is more than one x-intercept, separate them with commas. If there are no x-intercepts, click on "None".

x-intercept(s):

(d)

Find the equation of the axis of symmetry.

equationofaxisofsymmetry:

Explanation / Answer

The general form of the parabola is y = ax2 + bx+ c

From the given graph , we can observe that the curve cuts x - axis at x = 1 and x = 3

hence 0 = a(1)2 + b(1) + c implies that a + b + c = 0

and    0 = a(3)2 + b(3) + c implies that 9a + 3b + c = 0

By solving these two equations , we get 8a + 2b = 0 implies that 4a + b = 0 => b = - 4a

and we can observe that the curve cuts y - axis at y = - 3 hence , c = - 3

Therefore, from a + b + c = 0 we get a - 4a - 3 = 0 implies a = -1 and then b = 4

The equation of the given parabola is y = -x2 + 4x - 3

( a ) The curve is open downward.

( b ) The x-coordinate of the vertex is x = - b/(2a) = -4/(-2) = 2

Therefore, the x-coordinate of the vertex is x = 2.

( c ) Notice that the x-intercepts of any graph are points on the x-axis and therefore have y-coordinate 0. We can find these points by plugging 0 in for y and solving the resulting quadratic equation y = -x2 + 4x - 3

That is -x2 + 4x - 3 = 0 implies that x = 1 or x = 3

Hence , the x - intercepts are x = 1 and x = 3

( d )

Let's look at the quadratic equation for the parabola in the above Figure . The a-value is -1 and the b-value is 4. The equation of the axis of symmetry is:

x = -b/(2a) = -4/2(-1) = 4/2 = 2

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