Question

Maribel draws the graph of the function f(x)=2x^2−12x+14
. Complete the inequality to show the range of this function using the graph.

Answer 1

To find the range of the function \( f(x) = 2x^2 - 12x + 14 \), we first observe that it is a quadratic function in the standard form \( ax^2 + bx + c \), where:

- \( a = 2 \) (which is positive, indicating the parabola opens upwards),
- \( b = -12 \),
- \( c = 14 \).

To find the vertex of the parabola, we can use the vertex formula. The x-coordinate of the vertex for a quadratic function \( ax^2 + bx + c \) is given by:

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

Substituting in the values of \( a \) and \( b \):

\[
x = -\frac{-12}{2 \cdot 2} = \frac{12}{4} = 3
\]

Next, we find the corresponding y-coordinate (the function value at this x-coordinate) by substituting \( x = 3 \) back into the function:

\[
f(3) = 2(3)^2 - 12(3) + 14
\]
\[
= 2(9) - 36 + 14
\]
\[
= 18 - 36 + 14
\]
\[
= -18 + 14
\]
\[
= -4
\]

The vertex of the parabola is at the point \( (3, -4) \). Since the parabola opens upwards, the y-coordinate of the vertex, \( -4 \), is the minimum value of the function.

Thus, the range of the function \( f(x) \) can be expressed as:

\[
f(x) \geq -4
\]

Combining this with interval notation, the range of the function is:

\[
[-4, \infty)
\]

So, the complete inequality showing the range of the function is:

\[
f(x) \geq -4
\]