Problem 71 - Algebra and Trigonometry

Question

(a) On the same coordinate axes, sketch graphs (as accu- rately as possible) of the functions \(y=x^{3}-2 x^{2}-x+2 \quad\) and \(\quad y=-x^{2}+5 x+2\) (b) Based on your sketch in part (a), at how many points do the two graphs appear to intersect? (c) Find the coordinates of all intersection points.

Step-by-Step Solution

Answer

The graphs intersect at 3 distinct points: \((0, 2), (3, 8)\), and \((-2, -12)\).

Step 1: Understanding Function Behavior

Before plotting the graphs, let's understand the characteristics of each function. The first function is a cubic function, \( y = x^{3} - 2x^{2} - x + 2 \). Cubic functions can have 0 to 3 real roots and tend to have an 'S' shaped curve. The second function is a quadratic function, \( y = -x^{2} + 5x + 2 \). Quadratic functions have a parabolic shape, and this specific one opens downwards since the coefficient of \( x^2 \) is negative.

Step 2: Finding Key Points for Cubic Function

To sketch the cubic function \( y = x^{3} - 2x^{2} - x + 2 \), find its critical points by finding the derivative \( y' = 3x^2 - 4x - 1 \). Setting the derivative to zero will help identify turning points. Solving \( 3x^2 - 4x - 1 = 0 \), we use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This gives turning points which are crucial for the graph's shape.

Step 3: Sketching the Cubic Graph

Using the critical points found and evaluating the function at these points, plot these points on a graph. Also check where the cubic function crosses the x-axis by solving \( x^{3} - 2x^{2} - x + 2 = 0 \). Plot the y-intercept which is at \( (0, 2) \). Draw the curve smoothly connecting these points.

Step 4: Finding Key Points for Quadratic Function

For the quadratic function \( y = -x^2 + 5x + 2 \), identify its vertex using the formula \( x = -\frac{b}{2a} \) which gives \( x = \frac{5}{-2 \times -1} = 2.5 \). Compute \( y \) at this point to get the vertex. Also, find the roots by setting \( -x^2 + 5x + 2 = 0 \) and solving, either by factoring or using the quadratic formula.

Step 5: Sketching the Quadratic Graph

With the vertex, roots, and y-intercept \( (0, 2) \), sketch the parabola. Since it opens downward, draw the graph accordingly.

Step 6: Plotting Both Graphs on the Same Axes

On the same set of axes, plot both the cubic and quadratic functions using all the critical points, roots, y-intercepts, and the general behavior derived before. Use different colors or line styles for clarity.

Step 7: Identifying Points of Intersection

Visually inspect the graph to identify the points where the two graphs appear to intersect. These are potential solutions for part (b). Count how many intersections are visible on the graph.

Step 8: Solving for Intersection Points Algebraically

To find the exact intersection points, set the two equations equal: \( x^{3} - 2x^{2} - x + 2 = -x^{2} + 5x + 2 \). Simplify to \( x^3 - x^2 - 6x = 0 \) and solve for \( x \). Factor to get \( x(x^2 - x - 6) = 0 \) and solve for the roots. Calculate \( y \) for each \( x \).

Step 9: Finalizing Coordinates of Intersection Points

Substitute the x-values back into either function (preferably quadratic for simplification) to find the complete coordinate pairs for each intersection point.

Key Concepts

Cubic Function

A cubic function is characterized by its polynomial form which contains a term with a degree of three, represented as \( y = ax^3 + bx^2 + cx + d \). This gives it a distinctive "S" curve when plotted on a graph. One interesting aspect of cubic functions is their potential for having up to three real roots. These roots are the x-values where the function crosses the x-axis.
Understanding the properties of a cubic function can help pictorially represent its graph. Firstly, finding key points such as turning points can provide insights into the graph's shape. This is achieved by differentiating the function, \( y' = 3x^2 + 2bx + c \), and finding when this derivative equals zero. These points, where the slope of the tangent is zero, indicate maxima, minima, or points of inflection, offering a clearer picture of how the cubic function behaves throughout its domain.
Additionally, to get a better grasp of its features, consider the end behavior. For a positive leading coefficient, the graph will rise to the right and fall to the left. Conversely, with a negative leading coefficient, it will fall to the right and rise to the left. By combining the turning points and end-behavior analysis, you can effectively sketch the cubic function.

Quadratic Function

Quadratic functions are generally formulated as \( y = ax^2 + bx + c \), with their graph taking on a parabolic shape. The quadratic function you see here, \( y = -x^2 + 5x + 2 \), opens downward due to its negative leading coefficient. This transforms the parabola into an inverted "U" shape.
To start sketching, it's essential to identify the vertex of the parabola. The formula \( x = -\frac{b}{2a} \) helps you find the x-coordinate of the vertex. For this exercise, the vertex is found at \( x = 2.5 \). Substituting this back into the original equation gives the y-coordinate, helping locate this pinnacle point on the graph.
Alongside the vertex, determining the roots, where the function intersects the x-axis, adds structure to your sketch. These can be estimated by solving the quadratic equation \( -x^2 + 5x + 2 = 0 \) using the quadratic formula or factoring if possible. Roots and the vertex are paramount in constructing the accurate shape of the parabola.
Finally, finishing it with the y-intercept, where the graph crosses the y-axis, at \( (0, 2) \), shapes our understanding of the overall direction and presentation of the graph.

Graph Sketching

Sketching graphs may seem daunting, but it breaks down into manageable steps that highlight important aspects of the function. For exercises involving multiple functions, sketching on the same axes is useful for understanding their behavior and interactions.
Begin by plotting critical points for each function. For the cubic function, mark its roots, turning points, and y-intercept. For the quadratic, note its vertex, roots, and y-intercept. The purpose is to map out key characteristic points, facilitating a smoother drawing process as you connect these points.
Using these plotted points, draw the curves of the functions. Ensure the cubic graph displays its characteristic "S" curve and the quadratic its inverted parabola shape. It's important to visually distinguish these curves, possibly using different colors or styles for clarity when they are plotted on the same axis.
Once each function is correctly sketched, look for points where the graphs intersect. These intersections help solve for the coordinates where the functions equate. For some problems, algebraic solutions may be necessary, but visual sketching provides a critical first look at potential points.
This methodological approach to graph sketching unravels the intricacies of function behavior, promotes a deeper comprehension of mathematical concepts, and prepares students for complex intersection problems.