Axis summary

Question

 

Write the equation of the axis of symmetry, and find the coordinates of the
vertex of the graph of each function. Identify the vertex as a maximum or
minimum. Then graph the function.

y = - x ^2 + 2x + 3

 

 

Answer 1

y = ax² + bx + c , if a > 0 vertex is a minimum of quadratic function (or the lowest point on a parabola),
                          if a < 0 vertex is a maximum of quadratic function (or the highest point on a parabola).
The x-coordinate of a vertex is x = -b/2a , and that will be an equation of the axis of symmetry. y-coordinate of the vertex is the maximum or minimum value of the function.
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y = -x² + 2x + 3 , a = -1, b = 2, c = 3. the x-coordinate of a vertex is x = -2/2(-1) = 1
To find the y-coordinate of the vertex or the maximum value of function need to substitute "x" by 1 in given equation y = (-1)·1² + 2·1 + 3 = 4 .
(1,4) - coordinates of vertex, x = 1 is equation of the axis of symmetry, y = 4 is the max of function.
Now, to build the graph we have to find the roots of equation -x² + 2x + 3 = 0 ,
x_1,2 = (-b ± √(b² - 4ac))/2a      x_1,2 = (-2 ± √(2² -4·(-2)·3)/(2·(-1)) , x₁ = (-2 + √28)/(-2) ~ (-1.65)
                                                                                                   x₂ = (-2 - √28)/(-2) ~ 3.65

Plot the coordinates of vertex, coordinate of y-intercept (0,3) then on x-axis mark the coordinates: -1.65 and 3.65 . Start drawing the left branch of parabola: down from vertex through (0,3) and -1.65 , after that right branch: down from vertex through 3.65.
P.S. You may begin to graph the function by making a table of values for some values of your function and then plot those values in a coordinate plane and draw a smooth curve through the points.