Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=-1.24 x^{2}+1.68 x$$

Step-by-Step Solution

Answer

Vertex: (0.68, 0.57); x-intercepts: (0, 0) and (1.35, 0).

Step 1: Understand the Format of the Function

The function given is of the form \(y = ax^2 + bx + c\). Here, \(a = -1.24\), \(b = 1.68\), and \(c = 0\). This is a quadratic function and will form a parabola when graphed.

Step 2: Identify Important Features of the Parabola

For a parabola \(y = ax^2 + bx + c\), the vertex \(x\)-coordinate is given by \(-\frac{b}{2a}\). We can use this to find the \(x\)-coordinate of the vertex.

Step 3: Calculate the x-coordinate of the Vertex

Insert \(a = -1.24\) and \(b = 1.68\) into the formula for the vertex. \[x = -\frac{1.68}{2(-1.24)} = 0.68 \approx 0.68\]

Step 4: Calculate the y-coordinate of the Vertex

Using \(x = 0.68\), substitute back into the original equation to find the \(y\)-coordinate of the vertex. \[ y = -1.24(0.68)^2 + 1.68(0.68) = -1.24(0.4624) + 1.68(0.68) = -0.573376 + 1.1424 = 0.569024 \approx 0.57 \]

Step 5: Find the x-intercepts

The \(x\)-intercepts occur where \(y = 0\). So, solve \(-1.24x^2 + 1.68x = 0\). Factor out \(x\): \[x(-1.24x + 1.68) = 0\] The solutions are \(x = 0\) or \[-1.24x + 1.68 = 0\] Solve for \(x\): \[ x = \frac{1.68}{1.24} = 1.35 \approx 1.35 \]

Step 6: Use a Graphing Calculator

Set up a graphing calculator with a viewing window that includes points around and between the \(x\)-intercepts and the vertex. Observe the graph for a visual confirmation of the vertex and intercepts. This confirms the coordinates calculated.

Key Concepts

Parabolas

In the world of mathematics, a parabola is a U-shaped curve that emerges from quadratic equations of the form \( y = ax^2 + bx + c \). The shape and orientation of a parabola are primarily determined by the coefficient \( a \).

• If \( a \) is positive, the parabola opens upwards, making a smiley face shape.
• If \( a \) is negative, like the one in our exercise with \( a = -1.24 \), the parabola opens downwards, looking like a frown.

Parabolas are symmetrical about a vertical line called the axis of symmetry, which passes through the parabola's vertex. For quadratic equations, parabolas plot out two primary features - the vertex and the x-intercepts, both crucial for understanding and graphing these functions.

Grasping these basic elements helps in visualizing the curve's position and relation to the coordinate axes. By using the equation's form and characteristics, we further understand how to deduce crucial points such as the vertex and intercepts, important for graph analysis.

Vertex Calculation

Calculating the vertex of a parabola is an essential step in understanding the behavior and graph of a quadratic function. The vertex is the highest or lowest point on a parabola depending on whether it opens downwards or upwards.

To find the vertex for a quadratic function given by \( y = ax^2 + bx + c \), we use the formula for the x-coordinate of the vertex:\[x = -\frac{b}{2a}\]Plugging in the values from our problem, with \( a = -1.24 \) and \( b = 1.68 \), we find:\[x = -\frac{1.68}{2(-1.24)} = 0.68\]This x-value of the vertex indicates not only the axis of symmetry but where the parabola turns in its direction due to the change in rate provided by the quadratic term.

Next, substituting \( x = 0.68 \) back into the function calculates the y-coordinate of the vertex:\[y = -1.24(0.68)^2 + 1.68(0.68) = 0.57\]Thus, the vertex is located at the coordinate \((0.68, 0.57)\). Understanding the vertex's calculation helps define the parabola's extremum point, ensuring the accuracy in plotting the curve.

X-Intercepts

X-intercepts are points where the graph of an equation crosses the x-axis. At these points, the value of \( y \) is zero, providing crucial insights into the solutions of quadratic functions.

To find the x-intercepts of our equation \( y = -1.24x^2 + 1.68x \), we solve for when \( y = 0 \):\[-1.24x^2 + 1.68x = 0\]This equation can be factorized by taking \( x \) common:\[x(-1.24x + 1.68) = 0\]This offers us solutions for \( x \), namely:

• \( x = 0 \): One intercept is at the origin.
• Solving \(-1.24x + 1.68 = 0\) gives \( x = 1.35 \).

Thus, the x-intercepts are \( x = 0 \) and \( x = 1.35 \). Each x-intercept provides specific points where the parabola crosses the x-axis, marking the roots of the equation. A graphing calculator is often beneficial to observe these intercepts in conjunction with other features like the vertex, completing the graph's visualization.