Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, and (c) verify your results algebraically. $$y=x^{2}+3 x-4$$

Step-by-Step Solution

Answer

The x-intercepts for the given equation \(y = x^{2} + 3x - 4\) are \(x=1\) and \(x=-4\).

Step 1: Graphing

Plot the function \(y=x^{2}+3x-4\) on a graphing utility. This can be achieved on popular platforms such as GeoGebra, Excel's graphing feature, Desmos, or others. The graph gives a visual idea about where the function touches or crosses the x-axis i.e., the x-intercepts.

Step 2: Estimate the X-intercepts

Examine the graph and identify the points where the line intersects or touches the x-axis. These points are also known as roots or zeros and their x-values are x-intercepts. On visual inspection, two x-intercepts can be seen.

Step 3: Verifierfy the X-intercepts Algebraically

To verify the x-intercepts found earlier, solve the equation \(x^{2}+3x-4=0\) which represent the x-coordinates of the points where y = 0. Solving quadratic equation can be done by factoring, completing the square, or using Quadratic Formula: \(-b\pm \sqrt{b^2 - 4ac}\over 2a \). In this case, the quadratic can be factored: \(x^2 + 3x - 4 = 0 \Rightarrow (x-1)(x+4) = 0\). Setting each factor equal to zero and solve for \(x\) gives \(x=1\) and \(x=-4\), which verifies the x-intercepts seen on the graph.

Key Concepts

x-intercepts

When graphing quadratic equations, it is important to understand the concept of x-intercepts. These are the points where the graph crosses the x-axis. To find x-intercepts, you need to determine the values of \( x \) for which the equation equals zero. This is because points on the x-axis have a \( y \)-coordinate of zero.

By examining the graph of the quadratic equation \( y = x^2 + 3x - 4 \), we observe the points where the curve intersects the x-axis. These x-values are the x-intercepts. In the case of our example, the x-intercepts appear at \( x = 1 \) and \( x = -4 \).

Identifying these intercepts graphically gives us a visual confirmation, but it's always a good idea to verify them algebraically for accuracy.

Quadratic Formula

The Quadratic Formula is a mathematical tool used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula is written as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula calculates the roots of the quadratic equation, which correspond to the x-intercepts of the quadratic function. It is especially useful when the quadratic cannot be easily factored. For the equation \( y = x^2 + 3x - 4 \), applying the formula would involve:

• \( a = 1, b = 3, c = -4 \)
• \( x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \)
• Solving this shows: \( x = \frac{-3 \pm 5}{2} \)
• Resulting in: \( x = 1 \) and \( x = -4 \)

This confirms the x-intercepts found on the graph and through factoring, illustrating the power and versatility of the Quadratic Formula in finding solutions.

Factoring Quadratic Equations

Factoring is a method that allows us to solve quadratic equations by expressing them as a product of linear factors. Not all quadratics are easily factored, but when they are, this method provides a quick and clear solution.

To factor a quadratic like \( x^2 + 3x - 4 = 0 \), you are looking to express it as \( (x - p)(x - q) = 0 \), where \( p \) and \( q \) are numbers that satisfy the equation individually. In our example, the expression can be factored as \( (x - 1)(x + 4) = 0 \).

After factoring, you set each factor equal to zero:

• \( x - 1 = 0 \) which gives \( x = 1 \)
• \( x + 4 = 0 \) which gives \( x = -4 \)

These solutions represent the x-intercepts, consistent with what we observed on the graph and by using the Quadratic Formula.