Problem 27 - A Graphical Approach to College Algebra

Question

Fill in each blank with the appropriate response. (Remember that the vertical stretch or shrink factor is positive.) The graph of \(y=-4 x^{2}\) can be obtained from the graph of \(y=x^{2}\) by vertically stretching by applying a factor of_______________ and reflecting across the _________-axis.

Step-by-Step Solution

Answer

Stretch by a factor of 4 and reflect across the x-axis.

Step 1: Identify the Original Function

The original function given is \( y = x^2 \), which is a standard parabola opening upwards.

Step 2: Transform the Vertical Stretch or Shrink

The function to transform into is \( y = -4x^2 \). Here, the coefficient of \( x^2 \) is \(-4\). The absolute value \(|-4| = 4\) indicates a vertical stretch by a factor of 4 from the original function \( y = x^2 \).

Step 3: Identify the Reflection Axis

The negative sign in \( y = -4x^2 \) indicates that the graph is reflected across the x-axis, as compared to the original \( y = x^2 \), which opens upwards.

Key Concepts

Vertical Stretch

A vertical stretch modifies the graph of a function by expanding it away from the x-axis, making it taller and narrower. In this scenario, we have the function transformation from \(y = x^2\) to \(y = -4x^2\). The coefficient in front of \(x^2\) in \(y = -4x^2\) is \(-4\), where we consider only the absolute value to determine the stretching factor. This means that the graph stretches vertically by a factor of 4.
To better understand, think of vertical stretch as multiplying the original y-values by a positive number greater than 1. In our case:

• The original y-value from \(y = x^2\) is multiplied by 4.
• Every point of the parabola is pulled 4 times further away from the x-axis than it was in the original function.

A vertical stretch simply makes the graph look taller and skinnier, as opposed to wide or flat. It's important to note that although the function \(-4x^2\) includes a negative, we ignore it when calculating the stretch factor as it relates to orientation, not stretching.

Reflection Across Axis

A reflection across an axis involves flipping the graph over a specific axis. In our example, the transformation from \(y = x^2\) to \(y = -4x^2\) includes a reflection across the x-axis. The reflection is indicated by the negative sign in front of the \(x^2\) term.To picture this, imagine holding the graph at the x-axis and flipping it over like a book page from top to bottom. The once upward opening parabola now flips to open downwards.Here’s how it works:

• The original parabola \(y = x^2\) which opens upwards, has all its positive y-values turned negative.
• For every point on the original curve, the y-coordinate becomes its opposite (e.g., +3 becomes -3).

This transformation causes the whole graph to have the opposite orientation in terms of y-values, hence reflecting across the x-axis.

Quadratic Function

Quadratic functions are polynomial functions of degree 2, commonly expressed in the form \( y = ax^2 + bx + c \). They produce parabolic graphs, which resemble a U-shape.The most basic quadratic function is \( y = x^2 \). It creates a parabola that opens upwards centered at the origin, which provides a starting point for many transformations.Key features of a quadratic function include:

• The vertex, which is the highest or lowest point on the graph. In \( y = x^2 \), the vertex is at (0,0).
• The axis of symmetry, which is a vertical line that passes through the vertex, keeping the parabola symmetric. For \( y = x^2 \), this line is x = 0.
• The direction of opening, which can be upwards if the coefficient of \(x^2\) is positive, or downwards if negative.

In transformations such as in the exercise, changes to a quadratic's equation affect these features. The negative sign and scaling factors not only affect the direction the graph opens but can also alter its height and width, demonstrating the power of quadratic transformation.