Problem 49 - 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. $$y=-4 x^{2}+4 x+3$$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y = -4x^{2} + 4x + 3\) can be visually approximated using a graph. However, for exact values, solving for \(x\) when \(y = 0\) algebraically will provide the precise x-intercepts.

Step 1: Graph the Equation

Input the function \(y = -4x^{2} + 4x + 3\) into a graphing calculator or graphing software. Observe the shape and points at which the graph crosses the x-axis.

Step 2: Approximate the X-Intercepts

Identify the points where the graph intersects the x-axis. These points are the x-intercepts of the graph. They may not be precise due to the constraints of visual inspection, but make a note of these values.

Step 3: Verify Results Algebraically

To algebraically find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\). 0 = -4\(x^2\) + 4x + 3. You can solve this quadratic equation using various methods like factoring, completing the square, or the quadratic formula \(x = [-b ± sqrt(b²-4ac)]/2a\). The roots are the x-intercepts. These should correspond to the approximate values obtained in step 2.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots (or solutions) of quadratic equations. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) represent constants, and \(x\) represents the variable for which we are solving.
The quadratic formula is expressed as:

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

This formula allows us to find the precise x-values where the function crosses the x-axis (x-intercepts).
The term \(b^2 - 4ac\) under the square root is called the discriminant. It determines the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root.
• If \(b^2 - 4ac < 0\), the roots are complex numbers.

For the equation \(-4x^2 + 4x + 3 = 0\), applying the quadratic formula precisely gives the x-intercepts, providing an algebraic verification of the graphical approach.

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis. At these points, the y-coordinate is zero. To find the x-intercepts of a quadratic function like \(-4x^2 + 4x + 3\), we set the equation to zero and solve for the variable \(x\).
When visually identifying x-intercepts on a graph, they represent the horizontal location of the intersections between the curve and the x-axis. However, relying solely on graphical methods might not always yield precise results due to the scale of the graph or display limitations.
This is why verifying x-intercepts algebraically, such as through the quadratic formula, is crucial. It ensures accurate calculation of these intercepts, supporting the approximate values initially observed graphically.

Graphing Utilities

Graphing utilities are digital tools, like graphing calculators or software programs, that help visualize mathematical functions. They plot equations on coordinate grids, allowing users to observe key features of a function.
By entering the function \(y = -4x^2 + 4x + 3\) into a graphing utility, students can:

• Quickly graph the equation and see its curve.
• Identify important graph features such as vertex and axis of symmetry.
• Approximate the x-intercepts where the function crosses the x-axis.

However, graphing utilities might not always capture exact values due to screen resolution. Therefore, while they are excellent for initial exploration, verifying results with analytical methods like the quadratic formula is essential for precise calculations. These utilities speed up the graphing process and enhance understanding by providing a visual reference.