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Question

A parabola is graphed on a coordinate plane with the x-axis ranging from negative 10 to 10 in increments of 1 and the y-axis ranging from negative 10 to 10 in increments of 1.

Maribel draws the graph of the function f(x)=2x2−12x+14
. Complete the inequality to show the range of this function using the graph.

(1 point)
The range of the function is y≥
.
Remaining Attempts : 3
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Answer 1

To find the range of the function \( f(x) = 2x^2 - 12x + 14 \), we can first rewrite the function in vertex form.

1. Start with the standard form of the quadratic: \[ f(x) = 2x^2 - 12x + 14 \]

2. Factor out the coefficient of \( x^2 \) from the first two terms: \[ f(x) = 2(x^2 - 6x) + 14 \]

3. Complete the square inside the parentheses. To do that, take half of the x coefficient (-6), square it, and add and subtract this value inside the parentheses: \[ -6 / 2 = -3 \quad \Rightarrow \quad (-3)^2 = 9 \] \[ f(x) = 2(x^2 - 6x + 9 - 9) + 14 \] \[ f(x) = 2((x-3)^2 - 9) + 14 \] \[ f(x) = 2(x-3)^2 - 18 + 14 \] \[ f(x) = 2(x-3)^2 - 4 \]

4. Now we have \( f(x) = 2(x-3)^2 - 4 \). The vertex of this parabola is at the point (3, -4). Since the coefficient of \( (x-3)^2 \) is positive (2), the parabola opens upwards.

5. The minimum value of the function (at the vertex) is \(-4\). Therefore, the range of the function is all values of \( y \) that are greater than or equal to \(-4\).

Thus, the completed inequality showing the range of the function is: \[ \text{The range of the function is } y \geq -4. \]