Problem 40 - Algebra 2

Question

Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=-3 x^{2}+12 x $$

Step-by-Step Solution

Answer

Vertex form: \(y = -3(x-2)^2 + 12\); Vertex: (2, 12); Axis: \(x = 2\); Opens downward.

Step 1: Identify the Standard Form

The given quadratic function is \( y = -3x^2 + 12x \). This is in the standard form of a quadratic, \( y = ax^2 + bx + c \), where \( a = -3 \), \( b = 12 \), and \( c = 0 \).

Step 2: Completing the Square to Find Vertex Form

We will complete the square to convert the function into vertex form, \( y = a(x-h)^2 + k \). Start by factoring out \(-3\) from the first two terms: \[y = -3(x^2 - 4x) \]

Step 3: Complete the Square Inside the Parentheses

To complete the square, add and subtract \((\frac{b}{2})^2\) inside the parentheses. Here, \(\frac{b}{2} = \frac{4}{2} = 2\), so we add and subtract \(2^2 = 4\): \[y = -3(x^2 - 4x + 4 - 4) \]\[y = -3((x^2 - 4x + 4) - 4) \]

Step 4: Simplify the Expression

Simplify the completed square part: \[y = -3((x-2)^2 - 4) \]

Step 5: Distribute and Write in Vertex Form

Distribute \(-3\) into the expression: \[y = -3(x-2)^2 + 12 \]This is the vertex form: \( y = a(x-h)^2 + k \) with \( h = 2 \), \( k = 12 \).

Step 6: Identify Vertex, Axis of Symmetry, and Direction of Opening

The vertex of the parabola is \((h, k) = (2, 12)\). The axis of symmetry is \(x = h = 2\). Since \(a = -3\) is negative, the parabola opens downwards.

Key Concepts

Completing the Square

Completing the square is a method used to convert a quadratic equation from its standard form into the vertex form. This method enhances the understanding of the geometric features of the parabola represented by the quadratic equation.

Here's how it works:

• Identify the quadratic and linear coefficients in the equation. For example, take the equation \( y = -3x^2 + 12x \).
• Factor out any common factors from the terms with \( x \). Here, we factor out \(-3\) from \(-3x^2 + 12x \) to get \( -3(x^2 - 4x) \).
• To make it a perfect square trinomial inside the parentheses, find \( \left(\frac{b}{2}\right)^2 \). Here, \( b = -4 \), so \( \frac{-4}{2} = -2 \) and \( (-2)^2 = 4 \).
• Add and subtract this square value inside the parentheses to form \( -3(x^2 - 4x + 4 - 4) \), which simplifies to \( -3((x-2)^2 - 4) \).
• Simplify to get the equation in vertex form: \( y = -3(x-2)^2 + 12 \).

This process not only transforms the equation into vertex form, \( y = a(x-h)^2 + k \), but also makes it easier to identify the parabola's vertex \((h,k)\).

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of the parabola and divides it into two mirror-image halves. In the vertex form equation \( y = a(x-h)^2 + k \), the axis of symmetry is always the line \( x = h \).

For example, in the vertex form \( y = -3(x-2)^2 + 12 \), we find that \( h=2 \). Therefore, the axis of symmetry is the line \( x=2 \).

Understanding the axis of symmetry is crucial because:

• It helps locate the vertex easily.
• It provides insight into the reflection property of parabolas.
• It is essential for sketching the graph accurately.

Always remember, regardless of whether the parabola opens upwards or downwards, the axis of symmetry remains the line through \( x=h \).

Direction of Parabola Opening

The direction in which a parabola opens, either upwards or downwards, is determined by the sign of the coefficient \( a \) in both the standard and vertex form of a quadratic equation. This tells us how the parabola behaves when plotted on a graph.

Here's a simple way to understand it:

• If \( a > 0 \), the parabola opens upwards. This means it has a lowest point, or a minimum.
• If \( a < 0 \), as in our example with \( y = -3(x-2)^2 + 12 \) where \( a = -3 \), the parabola opens downwards. This denotes a highest point, or a maximum.

Knowing the direction of opening helps in:

• Predicting the range of the quadratic function.
• Understanding and characterizing the graph's extremum (maximum or minimum point).

This key concept is a stepping stone towards analyzing the complete behavior of quadratic functions in various mathematical scenarios.