Tap the blue points to see coordinates.
STEP 1:Find the x-intercepts
To find the x-intercepts solve, the equation $ \color{blue}{ x^7+2x^6+x^5+2x^4+2x^3-3x^2-3x+7 = 0 } $
The solution of this equation is:
$$ \begin{matrix}x = -1.7981 \end{matrix} $$(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^7+2x^6+x^5+2x^4+2x^3-3x^2-3x+7 } $, so:
$$ \text{Y inercept} = p(0) = 7 $$STEP 3:Find the end behavior
The end behavior of a polynomial is the same as the end behavior of a leading term.
$$ \lim_{x \to -\infty} \left( x^7+2x^6+x^5+2x^4+2x^3-3x^2-3x+7 \right) = \lim_{x \to -\infty} x^7 = \color{blue}{ -\infty } $$The graph starts in the lower-left corner.
$$ \lim_{x \to \infty} \left( x^7+2x^6+x^5+2x^4+2x^3-3x^2-3x+7 \right) = \lim_{x \to \infty} x^7 = \color{blue}{ \infty } $$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) $:
$$ p^{\prime} (x) = 7x^6+12x^5+5x^4+8x^3+6x^2-6x-3 $$The x coordinate of the turning points are located at the zeros of the first derivative
$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.6294 & x_2 = -0.4167 \end{matrix} $$(cleck here to see a explanation of how to solve the equation)
To find the y coordinates, substitute the above values into $ p(x) $
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.6294 } \Rightarrow p\left(0.6294\right) = \color{orangered}{ 4.9981 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.4167 } \Rightarrow p\left(-0.4167\right) = \color{orangered}{ 7.6405 }\end{aligned} $$So the turning points are:
$$ \begin{matrix} \left( 0.6294, 4.9981 \right) & \left( -0.4167, 7.6405 \right)\end{matrix} $$STEP 5:Find the inflection points
The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 42x^5+60x^4+20x^3+24x^2+12x-6 $.
The zeros of second derivative are
$$ \begin{matrix}x_1 = 0.2767 & x_2 = -0.7866 & x_3 = -1.1917 \end{matrix} $$Substitute the x values into $ p(x) $ to get y coordinates
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.2767 } \Rightarrow p\left(0.2767\right) = \color{orangered}{ 5.9971 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.7866 } \Rightarrow p\left(-0.7866\right) = \color{orangered}{ 7.2821 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.1917 } \Rightarrow p\left(-1.1917\right) = \color{orangered}{ 6.8752 }\end{aligned} $$So the inflection points are:
$$ \begin{matrix} \left( 0.2767, 5.9971 \right) & \left( -0.7866, 7.2821 \right) & \left( -1.1917, 6.8752 \right)\end{matrix} $$