Problem 53 - Algebra 1

Question

Use the following information. The number of recreational vehicles (RVs) sold in the United States from 1985 to 1991 can be modeled by \(N=-9.5 t^{2}+48.9 t+343.5,\) where \(N\) represents the number of vehicles sold (in thousands) and \(t\) represents the number of years since 1985. Sketch a graph of the model for positive values of \(x\) and \(y\).

Step-by-Step Solution

Answer

The graph of the equation \(N=-9.5 t^{2}+48.9 t+343.5\) is a downward-opening parabola. The vertex and axis of symmetry can be found using the coefficients of the equation, and the graph can be sketched by plotting the vertex and other key points derived from the equation.

Step 1: Identify the components of the quadratic equation

The equation provided is \(N=-9.5 t^{2}+48.9 t+343.5\). This is a quadratic equation in the format \(ax^{2} + bx + c\). In this equation, \(a=-9.5\), \(b=48.9\), and \(c=343.5\). The variable \(N\) represents the number of RVs sold and \(t\) is the number of years since 1985.

Step 2: Determine the vertex and axis of symmetry

The vertex of a parabola given in the form \(ax^{2} + bx + c\) is \((-b/2a , f(-b/2a))\). Computing this for the given function, it gives \(\left( 48.9/(-2*(-9.5)) , N(48.9/(-2*(-9.5))) \right)\). The vertex gives us the maximum point of the parabola. The axis of symmetry is \(x=-b/2a\), which gives a vertical line passing through the vertex and divides the parabola into two equal parts.

Step 3: Sketch the Parabola

Firstly, mark the vertex on a set of axes. For the x-values pick values less than and greater than the vertex x-coordinate and compute the corresponding y-values by substituting x-values in the equation. Get points on the parabola by drawing a smooth curve passing through those points including the vertex.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra, involving expressions that can be graphed as parabolas. These equations are in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents the unknown variable. Quadratic equations have several key characteristics that make them significant:

• The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• The solutions to the equation are known as the roots and they are the points where the parabola intersects the x-axis.
• Quadratics are symmetrical around a vertical line known as the axis of symmetry.

Understanding these properties helps in sketching the graph of a quadratic function, and in solving problems related to real-life situations like the number of RVs sold over a period.

Vertex

The vertex of a parabola is a pivotal point located at its maximum or minimum, depending on whether the parabola opens up or down. For an equation in the standard form \( ax^2 + bx + c \), the vertex can be found using the formula:

• \( x = -\frac{b}{2a} \)
• Calculate the corresponding \( y \) value by substituting \( x \) back into the equation to find \( f\left(-\frac{b}{2a}\right) \).

The vertex provides essential information about the parabola:

• It indicates the highest or lowest point on the parabola, representing either the maximum or minimum value of the function.
• For the equation \( N = -9.5t^2 + 48.9t + 343.5 \), solving for the vertex gives the maximum number of RVs sold, indicating a peak year for sales.

Understanding the vertex is crucial for interpreting the significance of the quadratic in various contexts, be it profit maximization or analysis of trends.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It's unique to each quadratic function and is found using the formula:

• \( x = -\frac{b}{2a} \)

The significance of the axis of symmetry includes:

• It helps in accurately sketching the parabola since each side is a reflection of the other.
• In real-world applications, like our RV sales example, it identifies the year around which sales peak and balance out over time.
• Knowing the axis of symmetry aids in predicting the behavior of the function, even without calculating the exact values.

Incorporating the concept of the axis of symmetry provides a complete picture of the quadratic's shape and helps in analyzing different aspects such as symmetry and balance in data modeling.