Sketch the graph of the polynomial function $$ p(x) = -8x^4+26x^3-8x^2-24x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -8x^4+26x^3-8x^2-24x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 2 & x_3 = -\dfrac{ 3 }{ 4 } & x_4 = 2 \end{matrix} $$

(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) = -8x^4+26x^3-8x^2-24x } $, 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.

$$ \lim_{x \to -\infty} \left( -8x^4+26x^3-8x^2-24x \right) = \lim_{x \to -\infty} -8x^4 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -8x^4+26x^3-8x^2-24x \right) = \lim_{x \to \infty} -8x^4 = \color{blue}{ -\infty } $$

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) $:

$$ p^{\prime} (x) = -32x^3+78x^2-16x-24 $$

The x coordinate of the turning points are located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 2 & x_2 = 0.869 & x_3 = -0.4315 \end{matrix} $$

(cleck here to see a explanation of how to solve the equation)

To find the y coordinates, substitute the above values into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 2 } \Rightarrow p\left(2\right) = \color{orangered}{ 0 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.869 } \Rightarrow p\left(0.869\right) = \color{orangered}{ -14.3973 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.4315 } \Rightarrow p\left(-0.4315\right) = \color{orangered}{ 6.5002 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 2, 0 \right) & \left( 0.869, -14.3973 \right) & \left( -0.4315, 6.5002 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -96x^2+156x-16 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.515 & x_2 = 0.11 \end{matrix} $$

Substitute the x values into $ p(x) $ to get y coordinates

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.515 } \Rightarrow p\left(1.515\right) = \color{orangered}{ -6.4576 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.11 } \Rightarrow p\left(0.11\right) = \color{orangered}{ -2.7037 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.515, -6.4576 \right) & \left( 0.11, -2.7037 \right)\end{matrix} $$

Graphing Polynomials