Tap the blue points to see coordinates.
STEP 1:Find the x-intercepts
To find the x-intercepts solve, the equation $ \color{blue}{ -x^4+9x^2 = 0 } $
The solutions of this equation are:
$$ \begin{matrix}x_1 = 0 & x_2 = 3 & x_3 = -3 \end{matrix} $$(you can use the step-by-step polynomial equation solver to see a detailed explanation of how to solve the equation)
STEP 2:Find the y-intercepts
To find the y-intercepts, substitute $ x = 0 $ into $ \color{blue}{ p(x) = -x^4+9x^2 } $, so:
$$ \text{Y inercept} = p(0) = 0 $$STEP 3:Find the end behavior
The end behavior of a polynomial is the same as the end behavior of a leading term.
$$ \lim_{x \to -\infty} \left( -x^4+9x^2 \right) = \lim_{x \to -\infty} -x^4 = \color{blue}{ -\infty } $$The graph starts in the lower-left corner.
$$ \lim_{x \to \infty} \left( -x^4+9x^2 \right) = \lim_{x \to \infty} -x^4 = \color{blue}{ -\infty } $$The graph ends in the lower-right corner.
STEP 4:Find the turning points
To determine the turning points, we need to find the first derivative of $ p(x) $:
$$ p^{\prime} (x) = -4x^3+18x $$The x coordinate of the turning points are located at the zeros of the first derivative
$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0 & x_2 = \dfrac{ 3 \sqrt{ 2}}{ 2 } & x_3 = -3 \dfrac{\sqrt{ 2 }}{ 2 } \end{matrix} $$(cleck here to see a explanation of how to solve the equation)
To find the y coordinates, substitute the above values into $ p(x) $
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0 } \Rightarrow p\left(0\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ \frac{ 3 \sqrt{ 2}}{ 2 } } \Rightarrow p\left(\frac{ 3 \sqrt{ 2}}{ 2 }\right) = \color{orangered}{ \frac{ 81 }{ 4 } }\\[1 em] \text{for } ~ x & = \color{blue}{ -3 \frac{\sqrt{ 2 }}{ 2 } } \Rightarrow p\left(-3 \frac{\sqrt{ 2 }}{ 2 }\right) = \color{orangered}{ \frac{ 81 }{ 4 } }\end{aligned} $$So the turning points are:
$$ \begin{matrix} \left( 0, 0 \right) & \left( \dfrac{ 3 \sqrt{ 2}}{ 2 }, \dfrac{ 81 }{ 4 } \right) & \left( -3 \dfrac{\sqrt{ 2 }}{ 2 }, \dfrac{ 81 }{ 4 } \right)\end{matrix} $$STEP 5:Find the inflection points
The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -12x^2+18 $.
The zeros of second derivative are
$$ \begin{matrix}x_1 = \dfrac{\sqrt{ 6 }}{ 2 } & x_2 = - \dfrac{\sqrt{ 6 }}{ 2 } \end{matrix} $$Substitute the x values into $ p(x) $ to get y coordinates
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ \frac{\sqrt{ 6 }}{ 2 } } \Rightarrow p\left(\frac{\sqrt{ 6 }}{ 2 }\right) = \color{orangered}{ \frac{ 45 }{ 4 } }\\[1 em] \text{for } ~ x & = \color{blue}{ - \frac{\sqrt{ 6 }}{ 2 } } \Rightarrow p\left(- \frac{\sqrt{ 6 }}{ 2 }\right) = \color{orangered}{ \frac{ 45 }{ 4 } }\end{aligned} $$So the inflection points are:
$$ \begin{matrix} \left( \dfrac{\sqrt{ 6 }}{ 2 }, \dfrac{ 45 }{ 4 } \right) & \left( - \dfrac{\sqrt{ 6 }}{ 2 }, \dfrac{ 45 }{ 4 } \right)\end{matrix} $$