What are the important points needed to graph #y= -x^2+2x+4#?

Answer 1

#x#-intercepts at #(1-sqrt5, 0)# and #(1+sqrt5, 0)#, #y#-intercept at #(0,4)# and a turning point at #(1,5)#.

So we have #y = -x^2 + 2x +4#, and usually the sorts of 'important' points that are standard for including on sketches of quadratics are axis intercepts and the turning points.
To find the #x#-intercept, simply let #y=0#, then: #-x^2 + 2x +4 = 0#

Next, we finish the square, which will aid in identifying the turning point.

#x^2 - 2x + 1# is the perfect square, then we subtract one again to maintain the equality: #-(x^2 - 2x + 1) + 1 +4 = 0# #:. -(x-1)^2 + 5 = 0#
This is the 'turning-point' form of the quadratic, so you can read your stationary point right off: #(1,5)# (alternatively you could differentiate and solve #y' = 0#).
Now just transpose the equation: #(x-1)^2 = 5# #:. x- 1 = +- sqrt5# #:. x = 1+-sqrt5#
The #y#-intercept is easy, When #x=0#, #y = 4#.

There you have it, then!

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Answer 2

To graph the equation y = -x^2 + 2x + 4, follow these steps:

  1. Identify the vertex of the parabola using the formula: (-b/2a, f(-b/2a)).
  2. Find the y-intercept by substituting x = 0 into the equation.
  3. Determine the x-intercepts by solving the equation -x^2 + 2x + 4 = 0.
  4. Plot the vertex, y-intercept, and x-intercepts on the coordinate plane.
  5. Sketch the parabola passing through these points.
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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.

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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