Problem 58 - Algebra for College Students

Question

Find the maximum or minimum value of \(y\) for each function. $$y=x^{2}-2 x+5$$

Step-by-Step Solution

Answer

The minimum value of \(y\) is 4.

Step 1: Identify the Type of Function

The given function is a quadratic function of the form \(y = ax^2 + bx + c\). For \(ax^2\) where \(a > 0\), the parabola opens upwards and has a minimum value. Here, \(a = 1\), \(b = -2\), and \(c = 5\), so the function has a minimum value.

Step 2: Use Vertex Formula to Find the x-coordinate

The vertex formula for the x-coordinate of a quadratic function \(y = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). Plugging in \(a = 1\) and \(b = -2\), \(x = -\frac{-2}{2*1} = 1\).

Step 3: Find the y-coordinate by Substituting the x-coordinate

Substitute \(x = 1\) back into the original function to find \(y\). So, \(y = (1)^2 - 2(1) + 5 = 1 - 2 + 5 = 4\).

Step 4: Conclude the Minimum Value

The minimum value of the function occurs at the vertex. Therefore, the minimum value is \(y = 4\).

Key Concepts

vertex formula

To find the vertex of a quadratic function, we use the vertex formula. The vertex formula for the x-coordinate of the vertex of a quadratic function given by \(y = ax^2 + bx + c\) is: \[ x = -\frac{b}{2a} \]. This formula helps us locate the x-coordinate of the point where the parabola either reaches its maximum or minimum value. For our problem, where \(a = 1\) and \(b = -2\), plugging these into the formula gives us: \[ x = -\frac{-2}{2 \cdot 1} = 1 \]. This means the vertex occurs at \(x = 1\). The vertex is a crucial point since it represents the highest or lowest point of the parabola depending on the direction it opens. Since \(a > 0\) in our function, the parabola opens upwards, indicating the vertex is the minimum point on the graph.

minimum value

The minimum value of a quadratic function occurs at its vertex when the parabola opens upwards (\(a > 0\)). To find this minimum value, we need to determine the y-coordinate of the vertex. Our function is \(y = x^2 - 2x + 5\). We already found that the x-coordinate of the vertex is \(x = 1\). Next, we substitute this back into the original function to find the corresponding y-coordinate. So we calculate: \[ y = 1^2 - 2 \cdot 1 + 5 = 1 - 2 + 5 = 4 \]. This means the minimum value of the function is \(y = 4\), occurring at \(x = 1\). The vertex \((1, 4)\) gives us the minimum value of the quadratic function. Distinguishing this minimum value is important for understanding the behavior of the function's graph.

coordinate substitution

Coordinate substitution is the process of plugging a known value of \(x\) or \(y\) back into the original equation to find the corresponding coordinate. In this problem, after using the vertex formula to find \(x = 1\), we need to find the y-coordinate by substituting \(x = 1\) into the function. This process is straightforward yet fundamental:

• Identify the original function: \(y = x^2 - 2x + 5\)
• Substitute \(x\) with 1: \(y = (1)^2 - 2 \cdot 1 + 5\)
• Perform the arithmetic operations: \(y = 1 - 2 + 5 = 4\)

After following these steps, we conclude that the function's minimum value is \(y = 4\). Coordinate substitution allows us to efficiently determine specific values of the function, validating our calculations and interpretations.