Problem 31 - Algebra 1: Concepts and Skills

Question

Find the coordinates of the vertex. Make a table of values, using \(x\) -values to the left and to the right of the vertex. $$ y=-4 x^{2}-4 x+8 $$

Step-by-Step Solution

Answer

The vertex of the given quadratic function is \( (0.5, 6) \). The table of values includes the points \((0, 8)\), \((0.5, 6)\), and \((1, 0)\).

Step 1: Convert the Function to Vertex Form

The standard form of the quadratic function is \(y = ax^{2} + bx + c\). Our function is \(y = -4x^{2} - 4x + 8\). The formula for the x-coordinate of the vertex is \(h = -\frac{b}{2a}\). So, calculating the x-coordinate of the vertex: \(h = -\frac{-4}{2*(-4)} = 0.5\). Then by substituting \(h\) back into the equation, we find the y-coordinate of the vertex, \(k = -4*(0.5)^{2} - 4*0.5 + 8 = 6\). So the vertex is \( (0.5, 6) \).

Step 2: Create a Table of Values

After finding the vertex, it's time to determine some points to the left and right of the vertex to fill in the table. Let's use \(x\)-values of 0 and 1 (to the left and right of the vertex respectively). Just plug these values into the equation to find their corresponding \(y\)-values: when \(x = 0, y = -4*(0)^{2} - 4*0 + 8 = 8\), and when \(x = 1, y = -4*(1)^{2} - 4*1 + 8 = 0\). The table of values is thus: \((0, 8)\), (vertex) \((0.5, 6)\), and \((1, 0)\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that helps describe the shape and position of the curve. It represents the highest or lowest point of the parabola, depending on whether the parabola opens upwards or downwards. In the case of the quadratic function given, \( y = -4x^2 - 4x + 8 \), the parabola opens downwards because the coefficient of \( x^2 \) is negative.

• The formula to find the x-coordinate of the vertex in the standard form \( y = ax^2 + bx + c \) is \( h = -\frac{b}{2a} \).
• For our function, with \( a = -4 \) and \( b = -4 \), the x-coordinate \( h = 0.5 \).
• Substituting this back into the equation gives the y-coordinate \( k = 6 \), hence the vertex is \((0.5, 6)\).

The vertex not only reveals if the parabola reaches a maximum or minimum but also provides insight into the symmetry of the graph. Understanding the vertex helps in sketching the parabola and analyzing its behavior.

Table of Values

A table of values lets us visualize and understand the graph of a quadratic function by plotting individual points. By choosing x-values on either side of the vertex, we can see the curvature of the parabola more clearly.

• Starting with the vertex \((0.5, 6)\), we choose x-values like 0 and 1.
• When \( x = 0 \), substituting into the equation gives \( y = 8 \).
• When \( x = 1 \), substituting yields \( y = 0 \).

The table of values becomes:

• \((0, 8)\)
• \((0.5, 6)\) - vertex
• \((1, 0)\)

This helps in plotting the graph, as these points illustrate the general u-shape of the parabola. The table isn't just about numbers; it helps us "see" the function.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function really simplifies understanding of the parabola's graph: it gives direct insight into the vertex and the direction of the parabola's opening. The vertex form is written as \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.

• If you know the vertex, converting from standard form \( y = ax^2 + bx + c \) to vertex form can make graphing easier.
• This form clearly shows the vertex \((h, k)\) and the "a" value tells whether the parabola opens up or down.

In our example, converting the given function to vertex form would confirm the vertex \((0.5, 6)\) and the direction. Using the vertex form is especially useful for quickly sketching graphs and comparing multiple quadratic functions to see their main features.