Math Problem Statement
range of 32x2 +147x+7
Solution
To find the range of the quadratic function , we first need to determine if it has a minimum or maximum value, and then find the corresponding values.
Step 1: Identify the Parabola's Orientation
The coefficient of is positive (), which means the parabola opens upward. Therefore, the function will have a minimum value at its vertex.
Step 2: Find the Vertex
The formula to find the x-coordinate of the vertex for a quadratic function is:
Here, and . Substituting these values into the formula:
Step 3: Find the Minimum Value of the Function
Now, substitute back into the function to find the y-coordinate of the vertex, which gives us the minimum value of the function.
Let's compute this value:
Thus, the minimum value of the function is approximately -162.
Step 4: State the Range
Since the parabola opens upwards, the range of the function is all values greater than or equal to the minimum value. Therefore, the range is:
Would you like further details on any step? Or do you have any additional questions?
Related Questions:
- How do you find the vertex of a quadratic function?
- How do you determine whether a quadratic opens upwards or downwards?
- How can you graph a quadratic function to visualize its range?
- What is the effect of changing the coefficients of a quadratic function on its range?
- How do you calculate the vertex form of a quadratic function?
Tip:
To quickly determine if a quadratic function opens upward or downward, check the sign of the coefficient in the term . If , the parabola opens upward; if , it opens downward.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Parabolas
Vertex
Range of a Function
Formulas
x_{vertex} = -b / 2a
f(x) = ax^2 + bx + c
Theorems
Vertex Theorem
Parabola Opening Direction
Suitable Grade Level
Grades 9-11