Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each of the quadratics, identify the \(x\) - and \(y\) -coordinates of the vertex and determine whether the vertex is the highest point on the curve or the lowest point on the curve. $$ y=-\frac{2 x^{2}+7}{4}+\frac{3 x-1}{3} $$

Step-by-Step Solution

Answer

The vertex of the quadratic equation is at the point (1, \frac{7}{6}). Since the coefficient of \(x^2\) is negative, the graph opens downward, and thus the vertex is the highest point on the graph.

Step 1: Rewrite the equation in the standard form

Rewrite the equation as \(y = ax^2+bx+c\) which is the standard form of a quadratic equation. So, \(y= -\frac{1}{2}x^{2} + x + \frac{2}{3}\)

Step 2: Find the x-coordinate of the vertex

Using the formula for the x-coordinate of a vertex \(h = -\frac{b}{2a}\), the coefficients for \(x^2\) and \(x\) here are \(a= -\frac{1}{2}\) and \(b=1\) respectively. So, \(h= -\frac{1}{2*-1/2} = 1\).

Step 3: Find the y-coordinate of the vertex

Substitute \(x = 1\) into the original equation to get the y-coordinate of the vertex. So, \(y = -\frac{1}{2}*(1)^2 + 1 + \frac{2}{3} = -\frac{1}{2} + 1 + \frac{2}{3} = \frac{7}{6}\).

Step 4: Check if the vertex is the highest or lowest point

The coefficient of \(x^2\) in the original quadratic equation is negative, so the parabola opens downwards. Therefore, the vertex is the highest point on the graph.

Key Concepts

Standard Form of Quadratic Equation

The standard form of a quadratic equation is fundamental in algebra. This form is given by \( y = ax^2 + bx + c \). It clearly represents the equation with three main parts:

• a is the coefficient of \( x^2 \), and it defines the width and opening direction of the parabola.
• b is the coefficient of \( x \), impacting the slope and position of the parabola.
• c is the constant term, determining the point where the parabola intersects the y-axis.

Understanding this structure is crucial as it allows us to easily manipulate the equation for various analyses. By identifying the values of a, b, and c, you can perform conversions and calculations, such as vertex determination and graphing the curve.
With practice, rewriting any quadratic equation into the standard form becomes a straightforward process, essential for further mathematical operations.

Vertex Formula

To find the vertex of a parabola, you need the vertex formula. This formula is given by:\[h = -\frac{b}{2a}\]where a and b are the coefficients from the standard form equation \( ax^2 + bx + c \).
The vertex of the parabola is the point \( (h, k) \), where h is the x-coordinate and k is the y-coordinate. Once you have h, plug it back into the original quadratic equation to find k. For the given problem:

• a is \(-\frac{1}{2}\), and b is \(1\).
• Using the vertex formula, you calculate h as \(1\).

Substitute this back into the equation, and you'll find the y-coordinate k as \(\frac{7}{6}\).
This process helps in accurately plotting the parabola and understanding the graph's highest or lowest point.

Parabola Opening Direction

A parabola can open upwards or downwards, which is determined by the coefficient a in the quadratic equation \( y = ax^2 + bx + c \).

• If a is positive, the parabola opens upwards, making the vertex the lowest point on the graph.
• If a is negative, the parabola opens downwards, making the vertex the highest point on the graph.

In our example, \( a = -\frac{1}{2} \), which is negative. This means the parabola opens downwards. Consequently, the vertex becomes the highest point on the curve.
Recognizing the parabola's direction is crucial for accurately interpreting and predicting the graph's behavior in relation to real-world phenomena.