Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=-\frac{1}{2}\left(x^{2}-6 x-7\right)\)

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=-\frac{1}{2}\left(x^{2}-6 x-7\right)\) are \(x=7, -1\), as obtained by both the graphing and algebraic methods.

Step 1: Graphing the function

The quadratic function can be graphed using a graphing utility. We could use online softwares available or graphing calculators. Plot the equation \(y=-\frac{1}{2}\left(x^{2}-6 x-7\right)\) on the graph. The x-intercepts are the x-values where the graph intersects the x-axis (i.e., where \(y=0\)).

Step 2: Find the x-intercepts from the graph

The x-intercepts are the points on the graph where the curve touches or crosses the x-axis. Note these points as they will be verified next.

Step 3: Solve the function algebraically

To find the x-intercepts algebraically, set the function equal to zero and solve for \(x\). 0=-\frac{1}{2}(x^{2}-6x-7). Multiply through by -2 to clear the fraction: 0=x^{2}-6x-7.

Step 4: Solve the Quadratic Equation

Next, solve the quadratic equation \(x^{2}-6x-7=0\). Factoring the quadratic gives \((x-7)(x+1)=0\). Setting these factors equal to zero we find \(x=7, -1\). These are the x-intercepts.

Step 5: Verify the x-intercepts

Finally, compare the x-intercepts of the algebraic solution with the x-intercepts obtained from the graph in Step 2. They should match, if they do it means the solution has been verified successfully.

Key Concepts

Finding X-Intercepts

Understanding how to find the x-intercepts of a quadratic function is critical in analyzing its graph. The x-intercepts, also known as zeros or roots, are points where the graph crosses the x-axis, implying that the y-value is zero at these points. To find them algebraically, set the quadratic equation to zero and solve for x. For example, with a quadratic function such as \(y = -\frac{1}{2}(x^2 - 6x - 7)\), you would start by eliminating any fractions to simplify the equation to \(0 = x^2 - 6x - 7\).

The next step is to factor the quadratic if possible or use the quadratic formula. In this example, the equation factors into \((x - 7)(x + 1) = 0\), yielding the solutions \(x = 7\) and \(x = -1\), which are our x-intercepts. As an exercise improvement advice, clearly understanding the factoring process or applying the quadratic formula when the equation cannot be easily factored is essential.

Solving Quadratic Equations

Solving quadratic equations is a foundational skill for graphing quadratics. There are multiple ways to solve a quadratic equation, namely factoring, completing the square, and using the quadratic formula. In the case of the equation \(x^2 - 6x - 7 = 0\), factoring was the chosen method. When equations are more complex and cannot be factored easily, the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) becomes a powerful tool. It is worth noting that understanding the discriminant, \(b^2 - 4ac\), from this formula can also tell us about the nature of the roots of the equation—whether they are real or complex.

For a robust understanding, students should practice solving quadratics with all three methods to ensure they can handle any quadratic equation they encounter.

Graphing Calculator Usage

A graphing calculator is an invaluable resource for students studying quadratic functions. To effectively use a graphing calculator, input the function exactly as it is presented. For the function \(y = -\frac{1}{2}(x^2 - 6x - 7)\), setting the window to include the predicted x-intercepts is important. Once the graph is displayed, the x-intercepts can be identified by looking for points where the curve crosses the x-axis. Additionally, most graphing calculators have a 'zero' or 'root' feature that can be used to find the x-intercepts more precisely.

Learning to use the different functionalities of a graphing calculator not only aids in visualizing the shape of the graph but also enhances understanding of the characteristics of quadratic functions.

Quadratic Function Verification

Verification of a quadratic function's x-intercepts is crucial for ensuring that the solutions derived algebraically corroborate with the graphed function. After determining the x-intercepts through graphing and solving the equation algebraically, compare the results. If they match, you can be confident in the accuracy of your solutions. This process of cross-verification serves as a means to double-check your work, reducing the likelihood of errors.

Always remember that verification is a step not to be overlooked as it helps consolidate the understanding of the concept and reinforces the connection between the algebraic and graphical perspectives of quadratic functions.