Problem 100 - College Algebra

Question

Use the graph of \(f(x)=x^{2}\) to graph \(g(x)=(x+3)^{2}+1\).

Step-by-Step Solution

Answer

The graph of the function \(g(x) = (x+3)^{2} + 1\) is a parabola that opens upwards, identical in shape to the graph of \(f(x) = x^{2}\), but shifted three units to the left and one unit up. Its vertex is at the point \(-3, 1\).

Step 1: Understanding the original function

Start with the original function, \(f(x) = x^{2}\). This describes a basic parabola opening upwards - it's a smooth, curved shape which reaches its minimum value (the vertex) at \(x = 0\), and then increases without limit as \(x\) moves away in either the positive or negative direction.

Step 2: Applying the horizontal shift

Next, apply the horizontal shift of 3 units to the left to get \((x+3)^{2}\). This moves every point on the original graph of \(f(x) = x^{2}\) three units to the left. This means the vertex of the parabola is now at the point \(-3, 0\).

Step 3: Applying the vertical shift

Then apply the vertical shift of 1 unit upward to reach \(g(x) = (x+3)^{2} + 1\). This moves every point on the graph of \((x+3)^{2}\) one unit up. So, the vertex of the parabola now sits at \(-3, 1\) - three units to the left of the y-axis, and one unit above the x-axis.

Step 4: Sketching the resultant graph

Finally, sketch the graph of \(g(x) = (x+3)^{2} + 1\). It should look identical to the graph of \(f(x) = x^{2}\), but shifted three units to the left and one unit up.

Key Concepts

Horizontal Shift

Imagine sliding a parabola along the x-axis without changing its shape—that's a horizontal shift. Consider our example function, \(f(x) = x^2\), which has a standard upright parabola shape centered on the y-axis. When we apply a horizontal shift to create \(g(x) = (x+3)^2\), we are, in essence, taking each point of \(f(x)\)'s graph and moving it 3 units to the left.

Why the left and not the right, you may wonder? It's due to the '+3' inside the parentheses; it's a bit counterintuitive, but adding a positive number inside the function's argument shifts the graph in the opposite direction! So for \(g(x) = (x-2)^2\), the shift would be to the right by 2 units. Horizontal shifts are a foundation concept in understanding transformations in quadratic functions.

Vertical Shift

Similarly, a vertical shift involves moving a parabola up or down along the y-axis. For the function \(g(x) = (x+3)^2 + 1\), once we've already applied our horizontal shift, the '+1' outside signifies a one-unit upward movement from our new starting position. Every point on the parabola rises in unison, like the graph is taking an elevator trip. This vertical shift doesn't change the parabola's width or direction, just its height.

Subtracting from the function would take the parabola downwards. For example, \(g(x) = (x+3)^2 - 4\) would mean a 4-unit plunge. Understanding vertical shifts allows you to position parabolas perfectly along the y-axis.

Parabola Vertex

The vertex of a parabola is its turning point, determining its lowest or highest value depending on the opening direction. In the standard form of a quadratic equation, \(y = a(x-h)^2 + k\), the vertex is at \((h, k)\).

When graphing \(g(x) = (x+3)^2 + 1\), we see that the horizontal and vertical shifts position the new vertex at \((-3, 1)\). It is three steps left and one step up from the origin, which is the vertex of our basic \(f(x) = x^2\) graph. Remember, the sign inside the parentheses is the opposite of the horizontal shift, while the sign outside dictates the vertical movement directly. The vertex is a vital reference point for sketching the correct shape and location of a parabola.

Quadratic Functions

Quadratic functions are the bread and butter of parabolas. They follow the general form of \(y = ax^2 + bx + c\), where the leading term \(ax^2\) dictates that the graph will be a parabola. When \(a > 0\), the parabola opens upward like a smile, and when \(a < 0\), it opens downward. The constants b and c affect the parabola's shape and position but not the fact that it's a parabola.

The exercise \(g(x) = (x+3)^2 + 1\) is a quadratic function that's been tweaked from its standard form for us to practice graphing shifts. These functions model a wide array of real-world situations, from projectile motion to profit maximization in business. Their versatility and visual nature make understanding quadratic functions and their transformations a key math skill.