Math Problem Statement

range of 32x2 +147x+7

Solution

To find the range of the quadratic function f(x)=32x2+147x+7f(x) = 32x^2 + 147x + 7, 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 x2x^2 is positive (3232), 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 ax2+bx+cax^2 + bx + c is:

xvertex=b2ax_{\text{vertex}} = \frac{-b}{2a}

Here, a=32a = 32 and b=147b = 147. Substituting these values into the formula:

xvertex=1472×32=147642.296875x_{\text{vertex}} = \frac{-147}{2 \times 32} = \frac{-147}{64} \approx -2.296875

Step 3: Find the Minimum Value of the Function

Now, substitute xvertexx_{\text{vertex}} back into the function to find the y-coordinate of the vertex, which gives us the minimum value of the function.

f(2.296875)=32(2.296875)2+147(2.296875)+7f(-2.296875) = 32(-2.296875)^2 + 147(-2.296875) + 7

Let's compute this value:

f(2.296875)32(5.275)147(2.296875)+7f(-2.296875) \approx 32(5.275) - 147(2.296875) + 7 f(2.296875)168.8337.8+7f(-2.296875) \approx 168.8 - 337.8 + 7 f(2.296875)162f(-2.296875) \approx -162

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:

[162,)\boxed{[-162, \infty)}


Would you like further details on any step? Or do you have any additional questions?

Related Questions:

  1. How do you find the vertex of a quadratic function?
  2. How do you determine whether a quadratic opens upwards or downwards?
  3. How can you graph a quadratic function to visualize its range?
  4. What is the effect of changing the coefficients of a quadratic function on its range?
  5. 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 aa in the term ax2ax^2. If a>0a > 0, the parabola opens upward; if a<0a < 0, 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