Sketch the graph of the polynomial function $$ p(x) = -x^4+4x^3-5x^2+3x-4 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -x^4+4x^3-5x^2+3x-4 = 0 } $

Since above equation has no solutions we conclude that
polynomial has no x-intecepts.

(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^4+4x^3-5x^2+3x-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.

$$ \lim_{x \to -\infty} \left( -x^4+4x^3-5x^2+3x-4 \right) = \lim_{x \to -\infty} -x^4 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -x^4+4x^3-5x^2+3x-4 \right) = \lim_{x \to \infty} -x^4 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

To determine the turning point, we need to find the first derivative of $ p(x) $:

$$ p^{\prime} (x) = -4x^3+12x^2-10x+3 $$

The x coordinate of the turning point is located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = 1.8847 \end{matrix} $$

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

To find the y coordinate, substitute the above value into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.8847 } \Rightarrow p\left(1.8847\right) = \color{orangered}{ -1.9452 }\end{aligned} $$

So the turning point is:

$$ \begin{matrix} \left( 1.8847, -1.9452 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -12x^2+24x-10 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.4082 & x_2 = 0.5918 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.4082 } \Rightarrow p\left(1.4082\right) = \color{orangered}{ -2.4529 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.5918 } \Rightarrow p\left(0.5918\right) = \color{orangered}{ -3.2694 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.4082, -2.4529 \right) & \left( 0.5918, -3.2694 \right)\end{matrix} $$

Graphing Polynomials