Question

A) What is the range of the function \(f\left(x\right)\ =5x^2-20x+12\)? (Hint: Graph the function in
your calculator). (1 point)
\(f\left(x\right)\ge\)

Answer 1

To find the range of the quadratic function \(f(x) = 5x^2 - 20x + 12\), we can start by rewriting it in vertex form. The vertex form of a quadratic function is given by the equation:

\[
f(x) = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction of the parabola (upward if \(a > 0\), and downward if \(a < 0\)).

### Step 1: Find the vertex

The formula for the x-coordinate of the vertex for a quadratic \( ax^2 + bx + c \) is:

\[
h = -\frac{b}{2a}
\]

For our function, \( a = 5 \) and \( b = -20 \). Plugging in these values:

\[
h = -\frac{-20}{2 \cdot 5} = \frac{20}{10} = 2
\]

Now we substitute \( h \) back into the function to find the y-coordinate \( k \):

\[
f(2) = 5(2)^2 - 20(2) + 12
\]
\[
f(2) = 5 \cdot 4 - 40 + 12
\]
\[
f(2) = 20 - 40 + 12
\]
\[
f(2) = -8
\]

### Step 2: Determine the range

Since this quadratic opens upwards (as the coefficient of \(x^2\), which is 5, is positive), the vertex represents the minimum point of the function. Thus, the minimum value of \(f(x)\) occurs at \(x = 2\) and is equal to \(-8\).

Therefore, the range of the function is:

\[
f(x) \ge -8
\]

### Final Answer

The range of the function \(f(x) = 5x^2 - 20x + 12\) is:

\[
f(x) \ge -8
\]