Tap the blue points to see coordinates.
STEP 1:Find the x-intercepts
To find the x-intercepts solve, the equation $ \color{blue}{ x^6+2x^5+x^4+7x^3+7x^2-x+4 = 0 } $
The solutions of this equation are:
$$ \begin{matrix}x_1 = -1.433 & x_2 = -2.2146 \end{matrix} $$(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^6+2x^5+x^4+7x^3+7x^2-x+4 } $, so:
$$ \text{Y inercept} = p(0) = 4 $$STEP 3:Find the end behavior
The end behavior of a polynomial is the same as the end behavior of a leading term.
$$ \lim_{x \to -\infty} \left( x^6+2x^5+x^4+7x^3+7x^2-x+4 \right) = \lim_{x \to -\infty} x^6 = \color{blue}{ \infty } $$The graph starts in the upper-left corner.
$$ \lim_{x \to \infty} \left( x^6+2x^5+x^4+7x^3+7x^2-x+4 \right) = \lim_{x \to \infty} x^6 = \color{blue}{ \infty } $$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(x) $:
$$ p^{\prime} (x) = 6x^5+10x^4+4x^3+21x^2+14x-1 $$The x coordinate of the turning points are located at the zeros of the first derivative
$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.065 & x_2 = -0.7294 & x_3 = -1.9255 \end{matrix} $$(cleck here to see a explanation of how to solve the equation)
To find the y coordinates, substitute the above values into $ p(x) $
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.065 } \Rightarrow p\left(0.065\right) = \color{orangered}{ 3.9665 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.7294 } \Rightarrow p\left(-0.7294\right) = \color{orangered}{ 5.7579 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.9255 } \Rightarrow p\left(-1.9255\right) = \color{orangered}{ -6.3198 }\end{aligned} $$So the turning points are:
$$ \begin{matrix} \left( 0.065, 3.9665 \right) & \left( -0.7294, 5.7579 \right) & \left( -1.9255, -6.3198 \right)\end{matrix} $$STEP 5:Find the inflection points
The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 30x^4+40x^3+12x^2+42x+14 $.
The zeros of second derivative are
$$ \begin{matrix}x_1 = -0.3385 & x_2 = -1.5371 \end{matrix} $$Substitute the x values into $ p(x) $ to get y coordinates
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.3385 } \Rightarrow p\left(-0.3385\right) = \color{orangered}{ 4.8749 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.5371 } \Rightarrow p\left(-1.5371\right) = \color{orangered}{ -1.7362 }\end{aligned} $$So the inflection points are:
$$ \begin{matrix} \left( -0.3385, 4.8749 \right) & \left( -1.5371, -1.7362 \right)\end{matrix} $$