Problem 42 - 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. $$ y=4 x^{2}+8 x-3 $$

Step-by-Step Solution

Answer

Vertex: (-1, -7), Axis of symmetry: x = -1, Opens upwards.

Step 1: Identify Original Form

The given quadratic function is in standard form: \( y = ax^2 + bx + c \), where \( a = 4 \), \( b = 8 \), and \( c = -3 \).

Step 2: Calculate the Vertex

To convert the function to vertex form, use the formula \( h = -\frac{b}{2a} \) to find the x-coordinate of the vertex. Substitute \( b = 8 \) and \( a = 4 \) into the formula to get \( h = -\frac{8}{2(4)} = -1 \).

Step 3: Find the y-coordinate of the Vertex

Substitute \( x = -1 \) into the original equation to find the y-coordinate of the vertex: \( y = 4(-1)^2 + 8(-1) - 3 = 4(1) - 8 - 3 = -7 \). Thus, the vertex is \((-1, -7)\).

Step 4: Write in Vertex Form

Using the vertex \((-1, -7)\), the vertex form is \( y = a(x - h)^2 + k \). Substitute \( a = 4 \), \( h = -1 \), and \( k = -7 \) to get \( y = 4(x + 1)^2 - 7 \).

Step 5: Identify the Axis of Symmetry

The axis of symmetry for the quadratic function is the line \( x = h \). So, the axis of symmetry is \( x = -1 \).

Step 6: Determine the Direction of Opening

Since \( a = 4 \), which is positive, the parabola opens upwards.

Key Concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to express equations because it highlights key characteristics such as the vertex, which is the point where the parabola changes direction. The vertex form of a quadratic function is given by:

• \( y = a(x-h)^2 + k \)

Here, \( (h, k) \) is the vertex of the parabola.
To convert from the standard form \( y = ax^2 + bx + c \) to vertex form, we follow these main steps:

• Calculate \( h = -\frac{b}{2a} \), this gives us the x-coordinate of the vertex.
• Find the y-coordinate by substituting \( x = h \) back into the original equation.
• Assign these values to \( h \) and \( k \) in the vertex form equation.

Applying this to the example \( y = 4x^2 + 8x - 3 \), we find the vertex \((-1, -7)\) and rewrite it as \( y = 4(x + 1)^2 - 7 \).
This transformation makes analyzing the vertex and other features straightforward.

Axis of Symmetry

The axis of symmetry is an essential feature of any quadratic function. It is a vertical line that divides the parabola into two mirror-image halves and passes through the vertex.

• The equation for the axis of symmetry is always \( x = h \), where \( h \) is the x-coordinate of the vertex.
• Thus, once the vertex is known, finding the axis of symmetry is straightforward.

In our example, since the vertex is at \((-1, -7)\), the axis of symmetry is \( x = -1 \).
This line is crucial because it offers insight into the structure of the parabola and simplifies the process of graphing the function.

Direction of Opening

The direction in which a parabola opens is determined by the sign of the coefficient \( a \) in the quadratic equation.

• If \( a > 0 \), then the parabola opens upwards, resembling a U-shape.
• If \( a < 0 \), it opens downwards, looking like an upside-down U.

In the given example, \( a = 4 \), which is positive. Therefore, the parabola opens upwards.
Understanding the direction of opening is critical when analyzing or sketching a quadratic function's graph. It helps predict how the values of \( y \) change as \( x \) moves away from the vertex.