Problem 50 - Algebra 1: Concepts and Skills

Question

In Exercises \(49-51\), sketch the graphs of the three functions in the same coordinate plane. Then describe how the three parabolas are similar to each other and how they are different. $$ \begin{aligned} &y=x^{2}+x+1\\\ &y=x^{2}+2 x+1\\\ &y=x^{2}+3 x+1 \end{aligned} $$

Step-by-Step Solution

Answer

All parabolas have the same shape due to the presence of the same quadratic term. They are, however, shifted horizontally due to different coefficients in the linear term.

Step 1 - Graph functions

Let's start by graphing the three quadratic functions below using any graphing tool.\n\n\(y=x^{2}+x+1\) \n\n\(y=x^{2}+2x+1\) \n\n\(y=x^{2}+3x+1\)

Step 2 - Compare Shapes

When comparing the shapes of the functions, all three functions should look similar because they all have the same quadratic term \(x^{2}\).

Step 3 - Compare Position

Now, let's compare the positions of the parabolas. All parabolas will have the same axis of symmetry. However, they will appear to shift horizontally according to the linear term, which is \(x\), \(2x\), and \(3x\) respectively. Thus, \(y=x^{2}+2x+1\) should be shifted rightwards compared to \(y=x^{2}+x+1\), and \(y=x^{2}+3x+1\) should further shift rightwards compared to the previous function.

Key Concepts

Parabola

A parabola is a U-shaped curve that appears when graphing quadratic functions. These functions are often written in the form of \(y = ax^2 + bx + c\). In the case of the equations provided (\(y = x^2 + x + 1\), \(y = x^2 + 2x + 1\), \(y = x^2 + 3x + 1\)), each represents a parabola.

• All these functions start with the same quadratic term, \(x^2\), meaning their parabolas open upwards.
• They have different linear terms (\(x\), \(2x\), and \(3x\)), which influence the position but not the shape.
• The vertical stretch of these parabolas is the same since their leading coefficient \(a\) remains 1.

Even without graphing, we can expect these patterns. Each core component plays a role in shaping how parabolas look and behave on a graph.

Axis of Symmetry

The axis of symmetry for a parabola is a vertical line that passes through its vertex, effectively dividing it into two mirror-image halves. The general formula for the axis of symmetry in a quadratic function \(ax^2 + bx + c\) is \(x = -\frac{b}{2a}\).

• For \(y = x^2 + x + 1\), \(b = 1\), and thus the axis is \(x = -\frac{1}{2}\).
• For \(y = x^2 + 2x + 1\), \(b = 2\), and the axis is \(x = -1\).
• For \(y = x^2 + 3x + 1\), \(b = 3\), and the axis is \(x = -\frac{3}{2}\).

The axis of symmetry changes as the linear coefficient \(b\) changes, resulting in parabolas that look as if they shift across the graph's horizontal plane. However, they all maintain their characteristic symmetrical nature about their unique line.

Horizontal Shift

A horizontal shift is observed when each parabola seems to move left or right from its original position. In our exercise, it is mainly due to the change in the linear term \(bx\) of the quadratic equations.

• \(y = x^2 + x + 1\) has its vertex on one vertical line dictated by its axis of symmetry.
• Changing to \(y = x^2 + 2x + 1\) means the parabola shifts horizontally, reflecting a movement due to the new symmetry axis \(x = -1\).
• Further adjustment to \(y = x^2 + 3x + 1\) moves the axis of symmetry to \(x = -\frac{3}{2}\), causing another shift in position.

This horizontal movement does not affect the shape of the parabola. However, it changes the visual representation of each graph relative to the previous one, elucidating how each parabola settles into its distinct position on the plane.