Problem 32 - Calculus with Concepts in Calculus

Question

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ x^{2}-4 x+y=5 $$

Step-by-Step Solution

Answer

Translate the parabola \( y = -x^2 \) to the vertex \( (2, 9) \) and sketch it.

Step 1: Complete the Square

To sketch the graph, we start by rewriting the equation in a form that makes it easier to graph. The given equation is \( x^2 - 4x + y = 5 \). We complete the square for the \( x \) terms. Start by taking the coefficient of \( x \), which is \(-4\), divide by 2, and square it: \((-4/2)^2 = 4\). Add and subtract this inside the equation: \( (x^2 - 4x + 4) - 4 + y = 5 \). This simplifies to \((x - 2)^2 + y = 9 \).

Step 2: Rearrange the Equation

Now, make \( y \) the subject of the equation: \( y = 9 - (x - 2)^2 \). This represents a parabola that opens downward, with its vertex at \((2, 9)\).

Step 3: Identify the Translations

The equation \( y = 9 - (x - 2)^2 \) is a vertical translation of the basic parabola \( y = -x^2 \). The vertex \( (2,9) \) indicates the parabola is translated right by 2 and up by 9.

Step 4: Sketch the Graph

Draw the basic axes, \(x\) and \(y\). To represent the translation, introduce new axes \(X\) and \(Y\) such that \( X = x - 2 \) and \( Y = y - 9 \). The new equation in terms of \(X\) and \(Y\) is \( Y = -X^2 \). Sketch the graph showing both sets of axes.

Key Concepts

Complete the Square

Completing the square is a method used to simplify a quadratic expression, which in turn makes it easier to graph a quadratic equation. In this process, we start with a quadratic expression, such as the terms involving \(x\) in the equation \(x^2 - 4x + y = 5\). The concept involves creating a perfect square trinomial. To do this:

• Take the linear coefficient of \(x\), which is \(-4\).
• Divide it by 2 to get \(-2\).
• Square the result to get \(4\).

This squared term, \(4\), helps us complete the square by turning the quadratic expression \(x^2 - 4x\) into \((x-2)^2 - 4\). By incorporating this in the equation, we transform the original expression into \((x - 2)^2 + y = 9\).
Completing the square is crucial as it enables us to identify the standard form of a parabola, facilitating graphing. It highlights the parabolic vertex, helping us understand how the graph shifts from its basic form.

Parabola Transformations

Once the quadratic expression is completed, transformations of the parabola become easier to identify. The transformation can help us see how the original graph \(y = -x^2\) changes shape or position.After completing the square, our equation becomes \(y = 9 - (x - 2)^2\), which hints at a vertical transformation due to the constant term.

• The graph of the parabola opens downward because of the negative sign in front of \((x - 2)^2\).
• The expression \((x - 2)^2\) indicates a horizontal shift to the right by 2 units.
• The positive 9 indicates a vertical shift upward by 9 units.

Through these transformations, the vertex of the parabola shifts from the origin \((0,0)\) to the new vertex at \((2, 9)\). Identifying these changes is essential for correctly sketching or understanding the resulting graph compared to the original.

Coordinate Translations

Coordinate translations involve moving every point of a graph a certain direction and distance. This step allows you to visualize how a function's graph shifts spatially across the plane.With our equation \(y = 9 - (x - 2)^2\), we are actually saying each point on the graph moves according to these rules:

• The whole graph shifts 2 units to the right, affected by \((x - 2)\).
• The graph also shifts 9 units upwards, as directed by the \(+9\).

This means our regular coordinate system needs adjusting. Introducing a new set of axes \(X = x - 2\) and \(Y = y - 9\) simplifies this visualization. The adjusted equation \(Y = -X^2\) indicates that the graph is now centered around these new axes.
This translation is an effective method to conceptualize and draw shifts in the function, maintaining the graph's shape but altering its position relative to the original axes.