Problem 2 - A Graphical Approach to College Algebra

Question

Solve each problem. Do not use a calculator. Find the maximum \(y\) -value on the graph of \(y=-2 x^{2}+8 x-5\)

Step-by-Step Solution

Answer

The maximum y-value is 3.

Step 1: Identify the Equation Type

The given function is a quadratic function of the form \(y = ax^2 + bx + c\). Here, \(a = -2\), \(b = 8\), and \(c = -5\). Quadratic functions form parabolas on the graph. Since \(a < 0\), the parabola opens downward, indicating it has a maximum point.

Step 2: Find the Vertex's x-coordinate

The vertex of a parabola given by \(y = ax^2 + bx + c\) can be found at \(x = -\frac{b}{2a}\). Substituting the values of \(b\) and \(a\) into the formula: \(x = -\frac{8}{2(-2)} = 2\).

Step 3: Calculate the Maximum y-value

Substitute \(x = 2\) into the original equation to find the maximum \(y\)-value: \[y = -2(2)^2 + 8(2) - 5\]. \(y = -2(4) + 16 - 5 = -8 + 16 - 5 = 3\).

Step 4: State the Maximum y-value

The maximum value of \(y\) on the graph is \(3\), which occurs at \(x = 2\).

Key Concepts

Vertex Form

The vertex form of a quadratic function provides a convenient way to determine the vertex of the graph. In vertex form, a quadratic function is expressed as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. If you look at a quadratic function in standard form, \( y = ax^2 + bx + c \), it can be converted into vertex form by completing the square or using the vertex formula. This is particularly useful because knowing the vertex is key in graphing the function and finding the maximum or minimum values.
To find the vertex formula, focus on the standard form \( y = ax^2 + bx + c \). The vertex's x-coordinate is calculated with \( x = -\frac{b}{2a} \), which gives us the point of symmetry for the parabola. Once you have the x-coordinate, you can substitute it back into the original equation to get the y-coordinate. This process determines the point on the graph that represents the vertex. For our given problem, \( a = -2 \) and \( b = 8 \), so the vertex's x-coordinate is \( x = 2 \). Substituting back into the equation provides the y-coordinate, giving the vertex as \((2, 3)\).
Understanding vertex form helps not just in computations but also in visually analyzing the quadratic function's behavior on a graph.

Parabola

When we talk about quadratic functions, we often mention the term "parabola." A parabola is the U-shaped curve that results from graphing a quadratic function. Depending on the coefficient in front of the \(x^2\) term, the parabola can open either upwards or downwards.

• If \(a > 0\), the parabola opens upward, resembling a valley.
• If \(a < 0\), as in our exercise with \(a = -2\), the parabola opens downward, resembling a mountain.

In geometry, the vertex of the parabola is the highest or lowest point of the curve. When the parabola opens downward, as in our function \( y = -2x^2 + 8x - 5 \), the vertex is the highest point on the graph. In such cases, the vertex reflects the maximum value the function can achieve, making it particularly interesting in optimization problems or physical scenarios where a highest point is of interest, like the peak of a projectile's path.
A parabola exhibits symmetry around a vertical line called the axis of symmetry. This line passes through the vertex and can be represented by \( x = -\frac{b}{2a} \). For our example, \( x = 2 \) is the axis of symmetry, meaning the parabola is mirrored around this line, confirming the vertex stands at \((2, 3)\).

Maximum Value

The maximum value of a quadratic function is an essential aspect when the parabola opens downward. Knowing this value can be crucial for various real-life applications.
In mathematical terms, the maximum value of a quadratic function occurs at the vertex when \(a < 0\). In the given example, \( y = -2x^2 + 8x - 5 \), the vertex \((2, 3)\) represents this maximum. Therefore, on the graph of the function, as \(x\) increases or decreases from \(2\), the value of \(y\) decreases, making \(3\) the highest point the graph reaches.
The process to determine this maximum value is straightforward. First, find the x-coordinate of the vertex, which is \( x = -\frac{b}{2a} = 2 \). Next, substitute this x-coordinate into the original function to calculate the y-coordinate. This results in \( y = 3 \), which is the maximum value of the function.
Understanding how to find the maximum value of a quadratic function is not only a pivotal math skill but also provides insights into many real-world phenomena, such as finding the maximum height reached by a thrown ball or the highest point of a suspension bridge.