Problem 22 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=-6 x^{2}-15 x $$

Step-by-Step Solution

Answer

The parabola opens downwards.

Step 1: Identify the equation

First, identify the given equation which is \(y=-6 x^{2}-15 x\)

Step 2: Identify the coefficient of \(x^2\)

Then, identify the coefficient of \(x^2\) in the given equation. In this case, it is -6.

Step 3: Determine the direction of the parabola

As the coefficient of \(x^2\) is negative (-6), the parabola will open downwards. If it was positive, the parabola would open upwards.

Key Concepts

Understanding Parabolas

A parabola is a U-shaped curve that is seen in the graph of a quadratic function. Every quadratic function is represented by an equation of the form \( y = ax^2 + bx + c \). The shape and direction of a parabola are primarily determined by its leading term, which includes the coefficient of \( x^2 \). A parabola can open upwards or downwards based on the sign of this coefficient.

• Upward-opening parabolas have a positive \( a \) value (e.g., \( a > 0 \)).
• Downward-opening parabolas have a negative \( a \) value (e.g., \( a < 0 \)).

The vertex of a parabola is its highest or lowest point. For an upward-opening parabola, the vertex is the minimum point, while for a downward-opening one, it is the maximum. Parabolas are symmetric, and their line of symmetry passes through the vertex. Understanding these basic features helps in predicting and interpreting the behavior of quadratic functions.

How Coefficients Influence Parabolas

Coefficients in a quadratic equation play a crucial role in shaping the parabola. In the standard form \( y = ax^2 + bx + c \), each coefficient has a specific impact:

• The coefficient \( a \) determines the direction (up or down) and the width of the parabola.
• The coefficient \( b \) affects the position of the vertex horizontally.
• The constant term \( c \) moves the parabola up or down the y-axis.

The width of a parabola is inversely related to the absolute value of \( a \). A larger absolute value of \( a \) results in a steeper parabola, while a smaller absolute value makes it wider. The sign of \( a \) dictates the parabola's direction; as observed in the provided equation \( y = -6x^2 - 15x \), the negative \( a \) value indicates a downward-opening curve. Understanding these roles helps in effectively graphing quadratics.

Steps for Graphing Quadratic Functions

Graphing a quadratic function involves a few core steps to visualize the parabola accurately. Consider the general form \( y = ax^2 + bx + c \):

• Identify the direction the parabola opens. As noted, if \( a < 0 \), it opens downward; if \( a > 0 \), it opens upward.
• Find the vertex, which is located at \( x = -\frac{b}{2a} \). Substitute this back into the equation to find the y-coordinate of the vertex.
• Plot the axis of symmetry, which is the vertical line that passes through the vertex.
• Calculate additional points by substituting x-values into the equation, ensuring the parabola's shape is captured accurately.

By following these steps, students can accurately draw parabolas, understanding not only their shape but the influence of coefficients on their graph. Practice with different quadratic equations strengthens familiarity with these concepts.