Problem 20 - Algebra 2

Question

What is the effect on the graph of the equation \(y=x^{2}+2\) when the equation is changed to \(y=3 x^{2}-5 ?\)

Step-by-Step Solution

Answer

The graph is vertically stretched by a factor of 3 and shifted downward by 7 units.

Step 1: Identify the transformation components

The original equation is given as \( y = x^2 + 2 \). The transformed equation is \( y = 3x^2 - 5 \). We need to compare the two equations to identify changes in the coefficients and constant terms.

Step 2: Analyze the changes in terms of the coefficient of \(x^2\)

In the original equation, the coefficient of \(x^2\) is 1, while in the transformed equation, the coefficient is 3. This represents a vertical stretch. Specifically, the parabola becomes narrower because the value stretches vertically by a factor of 3.

Step 3: Analyze the changes in terms of the constant term

The original equation has a constant term of +2, while the transformed equation has a constant term of -5. This represents a vertical shift. Specifically, the entire graph shifts downward by 7 units, from +2 to -5.

Step 4: Summarize the total transformation

The transformation from \( y = x^2 + 2 \) to \( y = 3x^2 - 5 \) includes a vertical stretch (making the parabola narrower) by a factor of 3 and a vertical shift downward by 7 units.

Key Concepts

Parabola

A parabola is a smooth, symmetrical curve, U-shaped, and can open either upwards or downwards. It is a graph of a quadratic function. The standard form of a quadratic equation is written as: \(y = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants. The term containing \(x^2\) is what gives a quadratic its parabolic shape. If \(a > 0\), the parabola opens upwards, whereas if \(a < 0\), it opens downwards.

• The vertex of the parabola is the peak or lowest point of the curve, depending on its opening direction.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.
• Each parabola is symmetric around its axis of symmetry.

Knowing these properties helps us understand transformations applied to the parabolic graph.

Vertical Stretch

A vertical stretch alters the width of a parabola by scaling it along the y-axis. When we change the coefficient of \(x^2\) from 1 to a number greater than 1, such as 3 in the equation \(y = 3x^2 - 5\), the parabola becomes narrower. Essentially, every point on the graph moves farther away from the x-axis.
The general effect of a vertical stretch:

• If the coefficient \(a\) increases from 1 to something bigger, the graph compresses and appears narrower.
• If \(a\) decreases and is less than 1 but greater than 0, the graph becomes wider, as it stretches vertically.

In essence, a vertical stretch by a factor of 3 means you multiply the original y-values by 3, for each point on the graph.

Vertical Shift

The vertical shift moves a graph up or down along the y-axis. When the constant term of a quadratic equation changes, it affects the vertical position of the entire graph. In our transformed equation \(y = 3x^2 - 5\), the constant moved from \(+2\) in the original equation to \(-5\), resulting in a downward vertical shift of 7 units.
Key points about vertical shifts:

• An increase in the constant term shifts the graph upward.
• A decrease shifts it downward.
• The shape of the graph itself remains unchanged; only its position along the y-axis is affected.

In this example, the graph shifted down by 7 units, lowering the vertex and all corresponding points on the parabola.