Problem 39 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Label the coordinates of the vertex. $$ y=4 x^{2}+8 x-3 $$

Step-by-Step Solution

Answer

The vertex of the function \( y=4x^{2}+8x-3 \) is (-1,-7). The function is a parabola opening upwards.

Step 1 - Complete the square

In order to find the vertex of the quadratic function \( y=4x^{2}+8x-3 \), convert this equation into vertex form by completing the square. Remember that \( x=h \) gives the x-coordinate of the vertex of the parabola. Starting with the given equation, rewrite it slightly to start completing the square, and group the terms with x in them: \( y=4(x^{2}+2x)-3 \). Now, add and subtract the square of half the coefficient of x inside the parentheses, then rearrange terms and factorise: \( y=4[(x+1)^{2}-1]-3 \) = \( 4(x+1)^{2}-4-3 \) = \( 4(x+1)^{2}-7 \).

Step 2 - Find the vertex

Now that we have the function in vertex form, we can find the vertex of the parabola. The vertex is given by the values (h,k). In this case, h is the value that makes the term in the square bracket zero and k is the constant term at the end. Here, \( h=-1 \) and \( k=-7 \). Thus, the vertex of the parabola is (-1,-7).

Step 3 - Sketch the graph

Since the coefficient of \( x^{2} \) is positive, the graph opens upwards and the vertex is a minimum point. Start by plotting the vertex (-1,-7). The axis of symmetry is the vertical line passing through the vertex, which is \( x=-1 \). Use the standard shape of a parabola as a guide and draw a curve passing through the vertex, opening upwards, symmetric about the line \( x=-1 \). Label the vertex point on the graph.

Key Concepts

Graphing Parabolas

The graph of a quadratic function is a U-shaped curve called a parabola. Graphing a parabola involves understanding its shape, direction, and position on the coordinate plane. Here are some essential elements when sketching parabolas:

• If the coefficient of the squared term is positive, the parabola opens upwards, forming a U-shape.
• If it's negative, the parabola opens downwards, creating a n-shape.
• The vertex of the parabola is either the highest or lowest point on the graph, depending on the direction it opens.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

When sketching, you typically start with the vertex and axis of symmetry. Then, you can use additional points or the shape of the parabola to complete your sketch.

Vertex Form

The vertex form of a quadratic equation is especially useful for quickly identifying the vertex of a parabola. It is expressed as: \[ y = a(x-h)^2 + k \] where:

• \(a\) determines the direction and the "width" of the parabola. If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
• \(h\) and \(k\) are the coordinates of the vertex, making it easier to graph the parabola.

Converting a quadratic equation to vertex form involves completing the square, which transforms the equation into one that clearly reveals the vertex. This form can make graphing and analyzing parabolas much easier.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into its vertex form. This method involves a few systematic steps:

• Start with a quadratic equation in standard form: \( ax^2 + bx + c \).
• Factor out the coefficient of \( x^2 \) (if it is not 1) from the terms involving \( x \).
• Add and subtract the square of half the coefficient of \( x \) inside the parentheses.
• Simplify and arrange the equation to achieve the vertex form, revealing the vertex of the parabola.

For example, to complete the square of \( y = 4x^2 + 8x - 3 \), you would transform it to \( y = 4(x + 1)^2 - 7 \). This adjustment presents the equation clearly in vertex form, identifying the vertex as \((-1, -7)\).

Vertex of a Parabola

The vertex is a crucial point on a parabola, as it indicates where the parabola changes direction. In the context of the equation \( y = 4(x+1)^2 - 7 \):

• The vertex is given by the point \((-1, -7)\). This corresponds to \( h = -1 \) and \( k = -7 \) in the vertex form equation \( y = a(x-h)^2 + k \).
• The vertex represents the minimum point of the parabola since the coefficient \( a = 4 \) is positive, indicating an upward opening.
• On a graph, this vertex would be the lowest point the parabola reaches.

By identifying the vertex, graphing the parabola becomes more intuitive, and you can quickly determine other characteristics like the axis of symmetry, which in this case is the vertical line \( x = -1 \).