State whether y=x^2 - 6x + 13 has maximum or a minimum value. Find the value.
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We have the quadratic function y equals x squared minus 6x plus 13. To determine if this function has a maximum or minimum value, we need to analyze its graph. Since this is a quadratic function with a positive coefficient for x squared, the parabola opens upward, which means it has a minimum value at its vertex.
Let's analyze the function step by step. First, we identify the coefficient of x squared, which is a equals 1. Since this coefficient is positive, the parabola opens upward. This tells us that the function has a minimum value, not a maximum value.
Now we need to find the vertex of the parabola. The x-coordinate of the vertex is given by the formula x equals negative b divided by 2a. In our function, a equals 1 and b equals negative 6. Substituting these values, we get x equals negative negative 6 divided by 2 times 1, which equals 6 divided by 2, which equals 3.
Now we substitute x equals 3 back into the original function to find the minimum value. y equals 3 squared minus 6 times 3 plus 13. This gives us y equals 9 minus 18 plus 13, which equals 4. Therefore, the minimum value of the function is 4, occurring at the point 3 comma 4.
To summarize: The quadratic function y equals x squared minus 6x plus 13 has a minimum value of 4. This occurs because the coefficient of x squared is positive, making the parabola open upward. The minimum value occurs at the vertex, which is located at x equals 3, giving us the point 3 comma 4.