Problem 26 - Algebra Form and Function

Question

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=18-2(x-2)^{2} $$

Step-by-Step Solution

Answer

Answer: The range of the function is \(y \leq 18\).

Step 1: Rewrite the given function in the form of a quadratic equation

First, let's rewrite the given function in a way that's easier to work with: $$ y = 18 - 2(x-2)^{2} $$ Expand the \((x-2)^2\) term: $$ y = 18 - 2(x^2 - 4x + 4) $$ Distribute the -2: $$ y = 18 - 2x^2 + 8x - 8 $$ Combine like terms: $$ y =-2x^2 + 8x + 10 $$ Now, the function is in the form of a quadratic equation \(y = ax^2 + bx + c\) with \(a = -2\), \(b = 8\), and \(c = 10\).

Step 2: Find the vertex of the quadratic equation

To find the vertex of the quadratic equation, use the formula: $$ x_{vertex} = \frac{-b}{2a} $$ Plug in the values for a and b: $$ x_{vertex} = \frac{-8}{2(-2)} = \frac{-8}{-4} = 2 $$ Now, we can find the y-coordinate of the vertex by plugging in \(x_{vertex}\) back into the equation: $$ y_{vertex} = -2(2)^2 + 8(2) + 10 = -2(4) + 16 + 10 = -8 + 16 + 10 = 18 $$ So, the vertex of the quadratic equation is \((2, 18)\).

Step 3: Determine the range

Since the coefficient of the \(x^2\) term is negative, the quadratic equation opens downwards. This means that the maximum value of the function occurs at the vertex. We already found that the y-coordinate of the vertex is 18. Therefore, the maximum value of the function is 18. The initial function \(y = 9 - (x-2)^2\) has the range \(y \leq 9\). For the given function, since the maximum value is 18, the range of the function will be: $$ y \leq 18 $$ Thus, the range of the function \(y = 18 - 2(x-2)^2\) is \(y \leq 18\).

Key Concepts

Range of a function

The range of a function is a set of all possible output values (y-values) that a function can produce. In simpler terms, it tells us how high or low the function can go as x changes. When we talk about quadratic functions, understanding the range is all about finding the highest or lowest point the function reaches.

In a quadratic function such as \[y = ax^2 + bx + c\]we need to look at the vertex for determining the range. If the parabola opens upwards (like a smile), the vertex will be the lowest point, determining the minimum value for y. If it opens downwards (like a frown), the vertex is the highest point, providing the maximum value for y.

• If \( a > 0 \), the parabola opens upwards, and the range is \( y \geq \text{minimum} \) value.
• If \( a < 0 \), the parabola opens downwards, and the range is \( y \leq \text{maximum} \) value.

In the function \( y = 18 - 2(x-2)^2 \), since \( a = -2 \) (a negative number), the parabola opens downwards. Therefore, the highest y-value, which occurs at the vertex, is the maximum value of 18. Hence, the range of this function is \( y \leq 18 \). That means y can be any number from negative infinity up to and including 18.

Vertex of a parabola

Understanding the vertex of a parabola is crucial in graphing and interpreting the function. The vertex represents either the maximum or minimum point of the parabola, depending on whether it opens upwards or downwards. For any quadratic function given by:\[y = ax^2 + bx + c\]the vertex can be calculated using the formula:
\[x_{vertex} = \frac{-b}{2a}\]Then, plug this x-value back into the original equation to find the y-coordinate.

In our exercise, we calculated the vertex of the function \( y = -2x^2 + 8x + 10 \). By substituting a = -2 and b = 8 into the formula, we find:\[x_{vertex} = \frac{-8}{2(-2)} = 2\]Next, substitute \( x = 2 \) back into the equation to find:\[y_{vertex} = -2(2)^2 + 8(2) + 10 = 18\]Thus, the vertex is at the point \((2, 18)\). This vertex tells us that the highest point, or maximum value of y, is 18 when \( x = 2 \). In real-world applications, finding the vertex helps identify optimal values of a function, such as maximum profit or minimum cost.

Graphing quadratics

Graphing quadratics is a fundamental skill that helps visualize how a quadratic equation behaves. The graph of a quadratic function is a parabola. It’s essential to understand some key features when graphing:

• Direction: Whether the parabola opens upwards or downwards is determined by the sign of the coefficient \(a\).
  • If \( a > 0 \), it opens upwards.
  • If \( a < 0 \), it opens downwards.
• Vertex: This is the key point on the graph which can be a maximum or a minimum.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images: \( x = x_{vertex} \).
• Y-intercept: The point where the graph crosses the y-axis, determined by setting \( x = 0 \).

Let’s consider the function \( y = -2x^2 + 8x + 10 \) again:- The parabola opens downwards because \( a = -2 \).- The vertex is at \((2, 18)\), so the axis of symmetry is \( x = 2 \).- To find the y-intercept, substitute \( x = 0 \) into the equation: \[y = -2(0)^2 + 8(0) + 10 = 10\]Thus, the y-intercept is at (0, 10). Graphing these points and features enables us to sketch an accurate graph of the quadratic function, showing how it rises or falls and its symmetry.