Problem 54 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE What is the axis of symmetry of the graph of \(y=x^{2}+3 x-2 ?\) $$ \begin{aligned} &A.)\quad x=-\frac{17}{4} &B.)\quad x=-\frac{3}{2} &C.)\quad x=\frac{3}{2} &D.)\quad x=\frac{19}{4} \end{aligned} $$

Step-by-Step Solution

Answer

The correct answer is C: \(x = \frac{3}{2}\)

Step 1: Identify 'a' and 'b'

In the equation \(y = x^{2} + 3x -2\), you can see that the coefficient of \(x^{2}\) is 1 (a = 1) and the coefficient of \(x\) is 3 (b = 3).

Step 2: Substitute values for 'a' and 'b' into the formula

To find the axis of symmetry, substitute a and b into the formula \(x = -b/2a\). If done correctly, it equates to \(x = -3/2(-1)\).

Step 3: Calculate the axis of symmetry

Performing the calculation \(-3/2(-1)\) gives \(x = 1.5\) or \(x = \frac{3}{2}\), which is the axis of symmetry.

Key Concepts

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). In this context, the coefficients \( a \), \( b \), and \( c \) are real numbers, with \( a eq 0 \). The simplest form of a quadratic equation is \( y = x^2 \), which when graphed forms a parabola. The general quadratic equation simulates this form, but includes adjustments that shift and reshape the parabola. Here are some key characteristics of quadratic equations:

• It is always a 'U' shaped curve when graphed, unless it is inverted to form an 'n'.
• The highest or lowest point of the quadratic is called the vertex.
• The line passing through the vertex and dividing the parabola into two mirror-image halves is the axis of symmetry.

To solve a quadratic equation, various methods can be used, like factoring, completing the square, or using the quadratic formula. Each method has its own unique procedure and application scenarios.

Graphing Parabolas

Graphing parabolas involves plotting a quadratic function on a coordinate plane. The parabola is the shape of the graph of a quadratic function. To graph a parabola, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation \( y = ax^2 + bx + c \).
• Calculate the axis of symmetry using the formula \( x = -\frac{b}{2a} \).
• Find the vertex, which lies on the axis of symmetry. It can be calculated by substituting the x-value of the axis of symmetry back into the equation to find the corresponding y-value.

The axis of symmetry is a vertical line that bisects the parabola into two equal halves. This line is crucial as it represents the x-value where the parabola's peak or trough occurs. Knowing the vertex and axis of symmetry helps in sketching the parabola accurately. For example, in the equation \( y = x^2 + 3x - 2 \), the axis of symmetry is found at \( x = \frac{3}{2} \), helping us understand that the vertex will lie along this line.

Vertex Form

The vertex form of a quadratic equation is a way of expressing the equation that highlights the vertex's location clearly. It is expressed as \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. Converting a quadratic in standard form (\( y = ax^2 + bx + c \)) to vertex form involves completing the square:

• The first step is to rearrange the equation and factor out \( a \) from the x terms, if necessary.
• Next, adjust the equation to create a perfect square trinomial, allowing you to express the quadratic as \( a(x-h)^2 + k \).
• This new expression highlights the vertex \( (h, k) \), making graphing simpler.

Using the vertex form is beneficial for quickly identifying shifts and transformations in the parabola's graph. It indicates whether the parabola faces upwards or downwards (depending on whether \( a \) is positive or negative) and how it shifts along the x and y axis based on \( h \) and \( k \). Familiarizing oneself with converting equations into vertex form aids in deeper understanding and graphing accuracy.