Problem 57
Question
Give a graph of the polynomial and label the coordinates of the intercepts, stationary points, and inflection points. Check your work with a graphing utility. $$ p(x)=x^{4}-2 x^{3}+2 x-1 $$
Step-by-Step Solution
Verified Answer
The y-intercept is at (0, -1). Use graphing tools for exact x-intercepts and stationary points.
1Step 1: Finding the Intercepts
To find the y-intercept of the polynomial \( p(x) = x^4 - 2x^3 + 2x - 1 \), substitute \( x = 0 \). This gives \( p(0) = -1 \), so the y-intercept is at the point \( (0, -1) \). For the x-intercepts, solve the equation \( x^4 - 2x^3 + 2x - 1 = 0 \) for \( x \). This is typically done using numerical methods or graphing as the polynomial does not factor easily.
2Step 2: Finding Stationary Points
Stationary points occur where the derivative is zero. First, find the derivative: \( p'(x) = 4x^3 - 6x^2 + 2 \). Set \( p'(x) = 0 \) to find points where the slope is zero. Using numerical methods, solve the equation \( 4x^3 - 6x^2 + 2 = 0 \). Approximate solutions might need a graphing tool.
3Step 3: Finding Inflection Points
Inflection points occur where the second derivative changes sign. First, find the second derivative: \( p''(x) = 12x^2 - 12x \). Set \( p''(x) = 0 \) and solve for \( x \) to determine potential inflection points: \( x(x-1) = 0 \), giving \( x = 0 \) and \( x = 1 \). Check if the second derivative changes sign around these values.
4Step 4: Sketching the Graph and Labeling Points
With the important points identified, sketch the polynomial. Indicate the intercepts, stationary points, and inflection points. The y-intercept is \((0, -1)\). Use a graphing utility to confirm the accuracy of the intercepts and shape, adjusting point approximations if needed.
Key Concepts
Polynomial InterceptsStationary PointsInflection Points
Polynomial Intercepts
Polynomials can have intercepts where the graph meets the axes. These are significant because they give us a quick look at where the polynomial has real roots or initial values.
The **y-intercept** occurs when the input, or x-value, is zero. To find this for the polynomial \( p(x) = x^4 - 2x^3 + 2x - 1 \), substitute \( x = 0 \). This results in \( p(0) = -1 \), so the y-intercept is at the point \( (0, -1) \). It tells you exactly where the curve crosses the y-axis.
The **x-intercepts** are where the graph meets the x-axis, which are crucial as they indicate the roots or solutions of the equation \( x^4 - 2x^3 + 2x - 1 = 0 \). Finding these requires solving the polynomial, which can be difficult. If simple algebraic manipulation doesn’t work, numerical methods or graphing calculators can help. Graphing helps visualize these points even when exact values are complex or irrational.
The **y-intercept** occurs when the input, or x-value, is zero. To find this for the polynomial \( p(x) = x^4 - 2x^3 + 2x - 1 \), substitute \( x = 0 \). This results in \( p(0) = -1 \), so the y-intercept is at the point \( (0, -1) \). It tells you exactly where the curve crosses the y-axis.
The **x-intercepts** are where the graph meets the x-axis, which are crucial as they indicate the roots or solutions of the equation \( x^4 - 2x^3 + 2x - 1 = 0 \). Finding these requires solving the polynomial, which can be difficult. If simple algebraic manipulation doesn’t work, numerical methods or graphing calculators can help. Graphing helps visualize these points even when exact values are complex or irrational.
Stationary Points
Stationary points on a polynomial graph are where the slope or gradient temporarily stops changing, and can become crucial points of interest in understanding the graph's behavior.
To find these, calculate the first derivative of the polynomial. For \( p(x) = x^4 - 2x^3 + 2x - 1 \), the derivative is \( p'(x) = 4x^3 - 6x^2 + 2 \). Set \( p'(x) = 0 \) to locate where the slope is zero. These are the x-values where the curve levels out and either peaks, troughs, or simply pauses on a straight path.
Finding the exact solutions might be challenging due to complex equations, so graphing tools are used for approximation. These points help in identifying the "turning points" of the polynomial, where it can change direction.
To find these, calculate the first derivative of the polynomial. For \( p(x) = x^4 - 2x^3 + 2x - 1 \), the derivative is \( p'(x) = 4x^3 - 6x^2 + 2 \). Set \( p'(x) = 0 \) to locate where the slope is zero. These are the x-values where the curve levels out and either peaks, troughs, or simply pauses on a straight path.
Finding the exact solutions might be challenging due to complex equations, so graphing tools are used for approximation. These points help in identifying the "turning points" of the polynomial, where it can change direction.
Inflection Points
Inflection points mark where the curve changes concavity – from being concave upwards (caved-up like a cup) to concave downwards (arched like a hill), or vice versa. These points help in understanding the overall geometry of the polynomial curve.
To find inflection points, assess where the second derivative of the polynomial changes sign. Start by finding the second derivative of \( p(x) = x^4 - 2x^3 + 2x - 1 \). This gives \( p''(x) = 12x^2 - 12x \).
Solve the equation \( p''(x) = 0 \), which is simplified to \( 12x(x-1) = 0 \), providing the x-values of potential inflection points: \( x = 0 \) and \( x = 1 \). It’s vital to check whether actual sign changes occur around these points to confirm true inflection points. Graphing can help visualize these changes, highlighting transitions in the curve’s shape.
To find inflection points, assess where the second derivative of the polynomial changes sign. Start by finding the second derivative of \( p(x) = x^4 - 2x^3 + 2x - 1 \). This gives \( p''(x) = 12x^2 - 12x \).
Solve the equation \( p''(x) = 0 \), which is simplified to \( 12x(x-1) = 0 \), providing the x-values of potential inflection points: \( x = 0 \) and \( x = 1 \). It’s vital to check whether actual sign changes occur around these points to confirm true inflection points. Graphing can help visualize these changes, highlighting transitions in the curve’s shape.
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