Question

(a) Graph fand \(g\) on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=-x^{2}+5 x ; \quad g(x)=x^{2}+3 x-4\)

Short answer

Graph both functions, solve the quadratic equation for points of intersection, then shade the region where \(f(x) > g(x)\).

Step-by-step solution

- Graph both functions

We start by graphing the functions on the same Cartesian plane. The equations are given as \(f(x) = -x^2 + 5x\) and \(g(x) = x^2 + 3x - 4\). Create a table of values for both functions to plot points and then draw the curves for the two equations.

- Find the points of intersection

To solve \(f(x) = g(x)\), set \(-x^2 + 5x = x^2 + 3x - 4\). Rearrange and solve for \(x\) by combining like terms and solving the resulting quadratic equation.

- Solve the equation

Combine the equations to get \(-x^2 + 5x = x^2 + 3x - 4\). Rearrange to form a single equation: \(-x^2 + 5x - x^2 - 3x + 4 = 0\), which simplifies to \(-2x^2 + 2x + 4 = 0\). Simplify further to get \-x^2 + x + 2 = 0\. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = -2, b = 2, c = 4\).

- Calculate the points of intersection

Using the quadratic formula, calculate the solutions for \(x\). We get \(x_1\) and \(x_2\) as the solutions. Then, substitute these \(x\) values back into either function to find the corresponding \(y\)-coordinates. These are the points of intersection.

- Label the points of intersection

Use the solutions from Step 4 to label the points of intersection on the graph made in Step 1. These points are where the two graphs cross.

- Shade the region for which \(f(x) > g(x)\)

Identify the regions on the graph where \(f(x) > g(x)\). Shade these regions which are below the curve of \(f(x)\) and above the curve of \(g(x)\) between the points of intersection.

Key concepts

Graphing Quadratic Equations

To graph a quadratic equation, we start by understanding its basic form: a parabola. Quadratic equations are typically written in the form \(ax^2 + bx + c = 0\). For our problem, we have two functions: \(f(x) = -x^2 + 5x\) and \(g(x) = x^2 + 3x - 4\).
We can create a table of values for these functions to plot points and draw their curves. The function \(f(x) = -x^2 + 5x\) represents a downward-opening parabola due to the negative coefficient of \(x^2\). Alternatively, \(g(x) = x^2 + 3x - 4\) represents an upward-opening parabola because of the positive coefficient of \(x^2\).
Upon plotting the points on the Cartesian plane, draw smooth curves connecting the points to visualize the graphs.

Intersection of Curves

Finding the intersection points of \(f(x)\) and \(g(x)\) means solving \(f(x) = g(x)\). We begin by setting their equations equal: \( -x^2 + 5x = x^2 + 3x - 4\).
Rearrange this into a single quadratic equation: \(-x^2 + 5x - x^2 - 3x + 4 = 0\) simplifies to \(-2x^2 + 2x + 4 = 0\).
Next, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\). Here, \(a = -2\), \(b = 2\), and \(c = 4\).
After solving, we find the \(x\)-values at the points of intersection. Substitute them back into either \(f(x)\) or \(g(x)\) to get the corresponding \(y\)-coordinates. These coordinates pinpoint where the curves intersect.

Quadratic Inequalities

Quadratic inequalities involve expressions like \(f(x) > g(x)\), which translates to finding where one curve is above the other on the graph.
From our problem, we need to determine the region where \(f(x) > g(x)\). This region lies below the curve of \(f(x)\) and above the curve of \(g(x)\).
The points of intersection found earlier will serve as boundaries for this shaded region. By examining the graphs, we shade the area between these intersection points where \(f(x)\) dominates \(g(x)\). This visualization helps in understanding the inequality graphically.

Solving Quadratic Equations

We solve quadratic equations using various methods like factoring, completing the square, and the quadratic formula. For our exercise, we used the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
This formula calculates the \(x\)-values where the given quadratic equations intersect.
Given \(-2x^2 + 2x + 4 = 0\), identify \(a = -2\), \(b = 2\), and \(c = 4\). Plugging these into the quadratic formula, we solve for \(x\). There will be two solutions, \(x_1\) and \(x_2\).
Then, substitute these \(x\)-values back into the original equations to find their corresponding \(y\)-values, thus providing the intersection points \((x_1, y_1)\) and \((x_2, y_2)\). These calculations are crucial for accurate graphing and interpreting the results.