Question 705760
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Given the quadratic function *[tex \LARGE y\ =\ ax^2\ +\ bx\ +\ c]


First, determine the *[tex \LARGE x]-coordinate of the vertex.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x_v\ =\ \frac{-b}{2a}]


Then determine the *[tex \LARGE y]-coordinate of the vertex.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y_v\ =\ y(x_v)\ =\ a(x_v)^2\ +\ b(x_v)\ +\ c]


Then determine the *[tex \LARGE y]-coordinate of the *[tex \LARGE y]-intercept which is equal to:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y(0)\ =\ a(0)^2\ +\ b(0)\ +\ c\ =\ c]


Then set the function equal to zero and solve the resulting quadratic equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ +\ c\ =\ 0\ \Right\ x_1\ =\ \frac{-b\ +\ \sqrt{b^2\ -\ 4ac}}{2a},\ x_2\ =\ \frac{-b\ -\ \sqrt{b^2\ -\ 4ac}}{2a}]


Then determine whether the graph opens upwards or downwards.  If the lead coefficient, that is *[tex \LARGE a] is positive, the graph opens upward, otherwise it opens downward.


With the above information you can answer all of the questions:


1.  Vertex:  *[tex \LARGE \left(x_v,\,y_v\right)]


2.  Axis of symmetry:  *[tex \LARGE x\ =\ x_v]


3.  Intercepts:  *[tex \LARGE \left(0,\,c\right)], *[tex \LARGE \ \left(x_1,\,0\right)], and *[tex \LARGE \left(x_2,\,0\right)]


4.  Domain:  The domain of all polynomial functions is the set of real numbers.


5.  Range:  If the graph opens upward, then the range is *[tex \LARGE \{y\,\in\,\mathbb{R}|\,y\ \geq y_v\}].  If the graph opens downward, then the range is *[tex \LARGE \{y\,\in\,\mathbb{R}|\,y\ \leq y_v\}]


6.  Increasing interval:  if graph opens upward, the increasing interval is from *[tex \LARGE y_v] to *[tex \LARGE \infty].  If the graph opens downward, the increasing interval is from *[tex \LARGE -\infty] to *[tex \LARGE y_v].


7.  Decreasing interval:  reverse the answers to 6. above.


8.  Plot the vertex, the y-intercept, and the two x-intercepts.  Use symmetry to plot *[tex \LARGE \left(2x_v,c\right)].  Draw a smooth curve through the 5 plotted points.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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