Tap the blue points to see coordinates.
STEP 1:Find the x-intercepts
To find the x-intercepts solve, the equation $ \color{blue}{ 5x^4-7x^3-5x^2+5x+1 = 0 } $
The solutions of this equation are:
$$ \begin{matrix}x_1 = -0.1774 & x_2 = -0.8467 & x_3 = 0.842 & x_4 = 1.582 \end{matrix} $$(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) = 5x^4-7x^3-5x^2+5x+1 } $, so:
$$ \text{Y inercept} = p(0) = 1 $$STEP 3:Find the end behavior
The end behavior of a polynomial is the same as the end behavior of a leading term.
$$ \lim_{x \to -\infty} \left( 5x^4-7x^3-5x^2+5x+1 \right) = \lim_{x \to -\infty} 5x^4 = \color{blue}{ \infty } $$The graph starts in the upper-left corner.
$$ \lim_{x \to \infty} \left( 5x^4-7x^3-5x^2+5x+1 \right) = \lim_{x \to \infty} 5x^4 = \color{blue}{ \infty } $$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) $:
$$ p^{\prime} (x) = 20x^3-21x^2-10x+5 $$The x coordinate of the turning points are located at the zeros of the first derivative
$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.3376 & x_2 = -0.5751 & x_3 = 1.2875 \end{matrix} $$(cleck here to see a explanation of how to solve the equation)
To find the y coordinates, substitute the above values into $ p(x) $
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.3376 } \Rightarrow p\left(0.3376\right) = \color{orangered}{ 1.9137 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.5751 } \Rightarrow p\left(-0.5751\right) = \color{orangered}{ -1.6508 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.2875 } \Rightarrow p\left(1.2875\right) = \color{orangered}{ -2.0513 }\end{aligned} $$So the turning points are:
$$ \begin{matrix} \left( 0.3376, 1.9137 \right) & \left( -0.5751, -1.6508 \right) & \left( 1.2875, -2.0513 \right)\end{matrix} $$STEP 5:Find the inflection points
The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 60x^2-42x-10 $.
The zeros of second derivative are
$$ \begin{matrix}x_1 = 0.8877 & x_2 = -0.1877 \end{matrix} $$Substitute the x values into $ p(x) $ to get y coordinates
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.8877 } \Rightarrow p\left(0.8877\right) = \color{orangered}{ -0.2936 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.1877 } \Rightarrow p\left(-0.1877\right) = \color{orangered}{ -0.0624 }\end{aligned} $$So the inflection points are:
$$ \begin{matrix} \left( 0.8877, -0.2936 \right) & \left( -0.1877, -0.0624 \right)\end{matrix} $$