Tap the blue points to see coordinates.
STEP 1:Find the x-intercepts
To find the x-intercepts solve, the equation $ \color{blue}{ 4a^3+12a^2-112a = 0 } $
The solutions of this equation are:
$$ \begin{matrix}a_1 = 0 & a_2 = 4 & a_3 = -7 \end{matrix} $$(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 $ a = 0 $ into $ \color{blue}{ p(a) = 4a^3+12a^2-112a } $, 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.
$$ \lim_{x \to -\infty} \left( 4a^3+12a^2-112a \right) = \lim_{x \to -\infty} 4a^3 = \color{blue}{ -\infty } $$The graph starts in the lower-left corner.
$$ \lim_{x \to \infty} \left( 4a^3+12a^2-112a \right) = \lim_{x \to \infty} 4a^3 = \color{blue}{ \infty } $$The graph ends in the upper-right corner.
STEP 4:Find the turning points
To determine the turning points, we need to find the first derivative of $ p(a) $:
$$ p^{\prime} (x) = 12a^2+24a-112 $$The x coordinate of the turning points are located at the zeros of the first derivative
$$ p^{\prime} (a) = 0 $$ $$ \begin{matrix}a_1 = 2.2146 & a_2 = -4.2146 \end{matrix} $$(cleck here to see a explanation of how to solve the equation)
To find the y coordinates, substitute the above values into $ p(a) $
$$ \begin{aligned} \text{for } ~ a & = \color{blue}{ 2.2146 } \Rightarrow p\left(2.2146\right) = \color{orangered}{ -145.7362 }\\[1 em] \text{for } ~ a & = \color{blue}{ -4.2146 } \Rightarrow p\left(-4.2146\right) = \color{orangered}{ 385.7362 }\end{aligned} $$So the turning points are:
$$ \begin{matrix} \left( 2.2146, -145.7362 \right) & \left( -4.2146, 385.7362 \right)\end{matrix} $$STEP 5:Find the inflection points
The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 24a+24 $.
The zero of second derivative is
$$ \begin{matrix}a = -1 \end{matrix} $$Substitute the a value into $ p(a) $ to get y coordinates
$$ \begin{aligned} \text{for } ~ a & = \color{blue}{ -1 } \Rightarrow p\left(-1\right) = \color{orangered}{ 120 }\end{aligned} $$So the inflection point is:
$$ \begin{matrix} \left( -1, 120 \right)\end{matrix} $$