Problem 21 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=-3 x^{2}+24 x $$

Step-by-Step Solution

Answer

The parabola opens down.

Step 1: Identify the given equation

The given equation is \(y=-3 x^{2}+24 x\), which is similar to the canonical form of a parabola equation \(y=a x^{2}+b x+c\)

Step 2: Identify the coefficient 'a' in the equation

The coefficient 'a' in the given equation is -3

Step 3: Determine the direction the parabola opens

Given that the coefficient 'a' is negative (-3), the parabola opens downwards. If 'a' was positive, the parabola would open upwards.

Key Concepts

Quadratic Equations

When dealing with quadratic equations, you're engaging with expressions that fit the general form of \(y=ax^2+bx+c\). These equations represent a specific type of graph called a parabola in a coordinate plane. The most significant aspect of these equations lies in their coefficients, particularly 'a', which determines the parabola's direction, 'b', which affects the position of the vertex horizontally, and 'c', which moves the parabola up and down on the y-axis.

Understanding quadratic equations is crucial not only for graphing but also for solving a variety of mathematical problems, including those involving areas, optimal values, and projectile motions. The solutions, or 'roots', of these equations can tell us where the parabola intersects the x-axis. These intersection points are found using methods such as factoring, completing the square, or the quadratic formula.

Coefficients in Parabolas

The coefficients in a parabola's equation play a pivotal role in its graphical representation. In the equation \(y=ax^2+bx+c\), each coefficient influences the parabola uniquely:

• \(a\): Determines whether the parabola opens upwards (\(a>0\)) or downwards (\(a<0\)). This coefficient also affects the 'width' of the parabola; a larger absolute value of 'a' results in a 'narrower' parabola, while a smaller absolute value leads to a 'wider' spread.
• \(b\): Impacts the parabola's horizontal placement. The axis of symmetry can be derived from \(x=-\frac{b}{2a}\), an important feature for locating the vertex of the parabola.
• \(c\): Represents the y-intercept of the parabola. This is where the graph crosses the y-axis.

In our example \(y=-3x^2+24x\), the coefficient \(a=-3\), indicates a 'downward' opening and a relatively narrow spread due to its magnitude.

Graphing Parabolas

Graphing a parabola involves plotting its shape based on the quadratic equation. Several key features are used to sketch the graph accurately:

• The direction of the parabola, which is determined by the sign of coefficient 'a'. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, as in our example, it opens downwards.
• The vertex, the highest or lowest point on the graph, is a critical feature which can be found using the formula \( (x = -b/(2a), y = f(-b/(2a))) \). The vertex also represents the axis of symmetry.
• The y-intercept is found by setting \(x=0\) and solving for 'y'.
• The x-intercepts or 'roots' can be determined by setting \(y=0\) and solving the resulting quadratic equation.

By combining these elements, you can create an accurate sketch of the parabola's graph. For our example equation \(y=-3x^2+24x\), we've established it opens downward due to the negative 'a' coefficient. The next steps in graphing would involve finding the vertex, axis of symmetry, and x and y intercepts, thus fully illustrating the characteristics and shape of the parabola.