Problem 41 - Algebra Form and Function

Question

Find the range of \(f\) by finding the values of \(a\) for which \(f(x)=a\) has a solution. $$ f(x)=2(x+3)^{2} $$

Step-by-Step Solution

Answer

Answer: The range of the function \(f(x) = 2 (x + 3)^2\) is \(a \ge 0\), which means that the function has solutions for all values of \(a\) greater than or equal to 0.

Step 1: Identify the vertex of the parabola

The given function, \(f(x) = 2 (x + 3)^2\), is in vertex form as \(f(x) = a(x - h)^2 + k\). In our case, the vertex is \((h, k) = (-3, 0)\), meaning the minimum or maximum value, depending on the direction the parabola opens, occurs at \(x = -3\).

Step 2: Determine the direction of the parabola

Since the coefficient of the \((x + 3)^2\) term is positive, namely \(2\), the parabola will open upwards. An upwards-opening parabola will have a minimum (and no maximum) value.

Step 3: Find the minimum value of the function

We will now find the minimum value of the function, or \(k\), by evaluating the function at the vertex. In our case, the minimum occurs at \(x = -3\), so we plug this value into the function: $$ f(-3) = 2(-3 + 3)^2 = 2(0)^2 = 0 $$ Thus, the minimum value of the function is \(k = 0\).

Step 4: Determine the range of the function

Since the parabola opens upwards and has a minimum value of \(0\), it follows that the range of the function includes all values greater than or equal to \(0\). Formally, we can express the range as: $$ a \ge 0 $$ In conclusion, the range of the function \(f(x) = 2 (x + 3)^2\) is \(a \ge 0\), meaning it has solutions for all values \(a\) greater than or equal to \(0\).

Key Concepts

Parabola Vertex Form

The vertex form of a parabola is a convenient way to write a quadratic function, making it easier to identify the parabola's vertex, which is a key feature of its graph. A quadratic function in vertex form is written as: \[ f(x) = a(x - h)^2 + k \] Here, \(a\) is a constant that affects the parabola's width and direction. The point \((h, k)\) is the vertex. In our case, we have the function \( f(x) = 2(x + 3)^2 \), meaning the vertex is at \((-3, 0)\). This tells us that the parabola's graph reaches its minimum or maximum value at \(x = -3\). The vertex form is especially useful in graphing or transforming quadratics, as well as finding the range and identifying the nature (minimum or maximum) of the parabola.

Minimum Value of a Parabola

The minimum value of a parabola is crucial when analyzing its graph and understanding its range. For a parabola in vertex form \( f(x) = a(x - h)^2 + k \), the minimum value is directly affected by the vertex and the sign of \(a\).

• If \(a\) is positive, the parabola opens upwards and has a minimum value at the vertex.
• If \(a\) is negative, the parabola opens downwards and the vertex is a maximum value.

For our function \(f(x) = 2(x + 3)^2\), the coefficient \(a = 2\) is positive, so the parabola opens upwards. The minimum value occurs at the vertex, \((h, k) = (-3, 0)\), and so the minimum output value is \(0\). Evaluating the function at \(x = -3\) confirms this: \[ f(-3) = 2(-3 + 3)^2 = 2 \times 0^2 = 0 \] This minimum value indicates that the lowest point of the parabola is \(0\), setting the lower limit of the function's range.

Direction of a Parabola

Understanding the direction in which a parabola opens is essential for analyzing its graph. This direction is dictated by the sign of the coefficient in front of the squared term, \(a\), in vertex form.

• If \(a > 0\), the parabola opens upwards, resembling a smiling face.
• If \(a < 0\), the parabola opens downwards, resembling a frowning face.

For the function \(f(x) = 2(x + 3)^2\), \(a = 2\), which is positive. This means the parabola opens upwards and is oriented such that it extends from its vertex point \((-3, 0)\) towards positive infinity. Since it opens upwards, it indicates that the vertex is a minimum point, and therefore, there is no upper boundary to the values generated by the function. This helps in determining that all values \(a\) greater than or equal to \(0\) fall within the range of the function.