Problem 36 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y = -2x^2\) is a downward opening parabola with the vertex at the origin (0,0).

Step 1: Identify the function and its form

The given function \(y = -2x^2\) is a quadratic function in the form \(y=ax^2\), where \(a\) is the coefficient of the \(x^2\), in this case \(a = -2\)

Step 2: Identify the vertex

For quadratic function in the form of \(y=ax^2\), the vertex is always at the origin (0,0). So, the vertex of this function is (0,0)

Step 3: Sketch the graph

Now we graph the function. As a is less than 0, the parabola opens downward. The vertex is at the origin (0,0), and as x moves away from 0 in either direction, y becomes more negative since we multiply x^2 (which is always positive) by -2.

Key Concepts

Graphing Parabolas

Understanding how to graph a parabola can seem challenging at first, but it’s quite simple when broken down into steps. Parabolas are the graphs of quadratic functions, which generally take the form \( y = ax^2 + bx + c \). For our specific function, \( y = -2x^2 \), it's in its simplest form, \( y = ax^2 \), where only the \( ax^2 \) component is present.

To graph a parabola:

• Identify the vertex (in the simplest form \( y = ax^2 \), it's at the origin (0,0) unless shifted).
• Determine the direction it opens; this depends on the sign of \( a \): a positive \( a \) opens upwards, while a negative \( a \) opens downwards.
• Plot the vertex and use additional points to map out the curve of the parabola. These can include easy-to-calculate points for small values of \( x \).

You then sketch the curve through these points, forming the characteristic "U" shape for upward-opening parabolas and an upside-down "U" for downward ones. Every quadratic function's parabola is symmetric around its vertical axis of symmetry. For our function, this axis is the \( y \)-axis, as the vertex is at the origin.

Vertex of a Parabola

The vertex of a parabola is one of its most defining features. It is essentially the peak or the lowest point on the graph, depending on whether the parabola opens upwards or downwards. In quadratic functions presented as \( y = ax^2 \), the vertex lies at the origin (0,0).

Here's a breakdown of the vertex:

• It represents the maximum or minimum value of the function.
• In \( y = ax^2 + bx + c \), you can find it using the formula \( x = -\frac{b}{2a} \) when both \( b \) and \( c \) are present.
• For \( y = -2x^2 \), the vertex is at (0,0), as \( b \) and \( c \) are absent, simplifying calculations.

The vertex gives you crucial information on the parabola's intercepts and curve direction. It ensures the graph accurately represents the function it visualizes by serving as the reference point for symmetry. To sketch accurately, always find and plot the vertex first.

Downward Opening Parabola

When working with parabolas, it is important to determine which way they open. For our function \( y = -2x^2 \), the parabola opens downward. This behavior is dictated by the coefficient \( a \), which is '-2' in this case. Here's how you can conclude this and what it means for the graph:

Signs and implications:

• The negative \( a \) value indicates that the parabola opens downwards. If \( a \) were positive, it would open upwards.
• This means the vertex is the highest point on the graph, as opposed to the lowest in upward-opening parabolas.
• As you move away from the vertex along the \( x \)-axis, the \( y \)-values decrease.

Visualizing a downward opening parabola, picture a frown. The farther away "x" moves from zero, the more sharply "y" decreases. This impacts the way the parabola looks and behaves, especially regarding its reach and shape, helping in understanding not just this particular graph, but others with similar characteristics.