Problem 23 - 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. $$ y=-x^{2}-4 x+8 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Standard Form

The given quadratic function is in the standard form as \( y = ax^2 + bx + c \). Here, \( a = -1 \), \( b = -4 \), and \( c = 8 \).

Step 2: Complete the Square

To rewrite the function in vertex form \( y = a(x-h)^2 + k \), we need to complete the square. Start by concentrating on the quadratic and linear terms: \( -x^2 - 4x \). Factor out the \( -1 \) from the x-terms to get \( -(x^2 + 4x) \).

Step 3: Form a Perfect Square Trinomial

To complete the square for \( x^2 + 4x \), take half of the coefficient of \( x \) (which is 4), divide by 2 to get 2, and then square it to get 4. Add and subtract 4 inside the parenthesis: \(-(x^2 + 4x + 4 - 4) = -(x^2 + 4x + 4) + 4\).

Step 4: Rewrite as a Perfect Square

The expression \( x^2 + 4x + 4 \) is a perfect square trinomial and can be written as \( (x + 2)^2 \). Therefore, rewrite the equation as \(y = -(x + 2)^2 + 8 + 4 = -(x + 2)^2 + 12\).

Step 5: Identify the Vertex

The function in vertex form is \( y = -(x + 2)^2 + 12 \). The vertex of the parabola is \( (h, k) = (-2, 12) \).

Step 6: Identify the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. For the vertex \((-2, 12)\), the axis of symmetry is \( x = -2 \).

Step 7: Determine the Direction of Opening

Since the coefficient of \( (x+2)^2 \) is \(-1\), which is negative, the parabola opens downwards.

Key Concepts

Vertex Form

The vertex form of a quadratic function is incredibly useful for analyzing the key features of a parabola. The vertex form is given by the equation: \[ y = a(x-h)^2 + k \] In this equation, \((h, k)\) represents the vertex of the parabola, and \(a\) indicates whether the parabola opens upwards or downwards. Writing a quadratic function in vertex form allows you to quickly pinpoint the vertex and understand the shape of the graph. - It provides an easy way to identify maximum or minimum values, which are found at the vertex. - Knowing the vertex allows you to also find the axis of symmetry. If a function is initially in standard form \(y = ax^2 + bx + c\), you can convert it to vertex form by completing the square, which we will discuss next.

Completing the Square

Completing the square is a powerful algebraic technique used to transform a quadratic function from standard form to vertex form. It involves creating a perfect square trinomial from the quadratic expression. For the equation \( y = -x^2 - 4x + 8 \), the process starts with isolating the quadratic and linear terms: - First, factor out the leading coefficient (even if it's negative) from the quadratic and linear terms: \[-(x^2 + 4x)\]. - Next, take half of the coefficient of \(x\) (which in this case is 4), divide it by 2 to get 2, and then square it to obtain 4. - Add and subtract this squared value within the parenthesis to form a perfect square trinomial: \[-(x^2 + 4x + 4 - 4)\]. - This can then be rewritten as \(-(x+2)^2 + 4\) to reflect the perfect square trinomial.Completing the square simplifies complex expressions and highlights key geometric properties of the graph.

Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line that passes through the vertex of the parabola. It divides the parabola into two mirror-image halves and can be easily identified once the function is in vertex form. Given the vertex form equation \( y = a(x-h)^2 + k \), the axis of symmetry is the line \( x = h \). For the function \( y = -(x + 2)^2 + 12 \), the vertex is at \((-2, 12)\), so the axis of symmetry is \( x = -2 \).- This line is crucial for graphing because it helps ensure accuracy with reflections. - It's also useful for determining the coordinates of points that share the same \(y\)-values on either side of the vertex.The axis of symmetry is a staple concept in analysis and graphing of quadratic functions.

Direction of Opening

The direction in which a parabola opens (upwards or downwards) depends on the sign of the coefficient \(a\) in the quadratic equation in vertex form. The function \( y = -(x + 2)^2 + 12 \) has \(a = -1\), which is negative.- When \(a\) is negative, the parabola opens downwards. - Conversely, if \(a\) were positive, the parabola would open upwards.Understanding the direction of opening is essential for identifying whether the vertex represents a maximum or a minimum point:- In a downward-opening parabola, like our example, the vertex represents the maximum point.- In an upward-opening parabola, the vertex represents the minimum point.This concept aids in predicting and describing the overall shape and behavior of the quadratic function's graph.