Question

Transform the equation \(\mathrm{X}^{3}-3 \mathrm{X}^{2}-\mathrm{Y}^{2}+3 \mathrm{X}+4 \mathrm{Y}-5=0 \mathrm{by}\) translating the coordinate axes to a new origin at \((1,2) .\) Plot the locus and show both sets of axes.

Short answer

The translated equation after shifting the coordinate axes to the new origin at \((1,2)\) is given by \(x^3 - y^2 - 2 = 0\). When plotted alongside the original equation \(X^3 - 3X^2 - Y^2 + 3X + 4Y - 5 = 0\), the translated curve is the original curve shifted to the new origin. Both sets of axes should be displayed on the graph.

Step-by-step solution

Translate the coordinate axes

To translate the coordinate axes to a new origin at \((1,2)\), we will replace \(X\) with \((x+1)\) and \(Y\) with \((y+2)\), where \((x,y)\) are the new coordinates and \((X,Y)\) are the original coordinates. The given equation is: \(X^3 - 3X^2 - Y^2 + 3X + 4Y -5 = 0\) Now, replace \(X\) with \((x+1)\), and \(Y\) with \((y+2)\): \((x+1)^3 - 3(x+1)^2 - (y+2)^2 +3(x+1) +4(y+2) -5 = 0\) Step 2: Simplify the translated equation

Simplify the translated equation

Now, we will simplify the equation by expanding the terms and combining like terms: \((x^3 + 3x^2 + 3x + 1) - 3(x^2 + 2x + 1) - (y^2 + 4y + 4) + 3x + 3 + 4y +8 - 5 = 0\) \(x^3 + 3x^2 + 3x + 1 - 3x^2 - 6x -3 - y^2 - 4y - 4 + 3x + 3 + 4y +8 - 5 = 0\) By combining like terms, we get: \(x^3 - y^2 - 2 = 0\) Step 3: Plot the curves

Plot the given curve and the translated curve

Now, we have the original equation and the translated equation we can plot: Original equation: \(X^3 - 3X^2 - Y^2 + 3X + 4Y - 5 = 0\) Translated equation: \(x^3 - y^2 - 2 = 0\) Use an appropriate graphing tool or software to plot both equations on the same graph, such as Desmos, Geogebra, or another tool of your choice. After plotting, notice that the curve described by the translated equation is indeed the curve described by the original equation shifted to the new origin at \((1,2)\). Make sure to show both sets of axes to complete this exercise.

Key concepts

Polynomial Functions

Polynomial functions are an essential branch of mathematics, and they appear frequently in various mathematical and real-world applications. A polynomial function is an expression consisting of variables and coefficients, fashioned into an equation of one or more terms. Each term in a polynomial is made up of a constant multiplied by a variable raised to a non-negative integer exponent.
For example, the equation given in your exercise is a polynomial:

• The term \(X^3\) represents a cubic polynomial, which indicates that the overall degree of the polynomial is 3.
• Other terms like \(-3X^2\), \(-Y^2\), and constants each contribute to defining the curve represented by the polynomial.

Understanding polynomial functions is crucial as they offer estimations for complicated real-world relationships, allowing us to graphically represent and analyze these relationships easily. Having a firm grasp of the behavior of different polynomial degrees helps in predicting how graph shapes change as you translate, rotate, or otherwise transform them.

Locus Plotting

Locus plotting involves tracing and plotting a set of points that satisfy specific conditions or equations. In this exercise, after translating the origin, the locus is described by the equation \(x^3 - y^2 - 2 = 0\). The set of all points \((x, y)\) that satisfy this equation make up the locus of the graph.Identifying the locus on a graph is useful because:

• It shows the collection of points that fulfill a certain mathematical condition.
• It illustrates how transformations, such as translations, shifts, and rotations, change the form of the curve but not the inherent relationship between \(x\) and \(y\).
• The locus helps visualize changes and dynamics within a system represented by the equation.

By plotting the locus pre- and post-translation, students can observe geometric transformations in practice, better understanding how equations direct the behavior of curves on a coordinate plane.

Graph Transformations

Graph transformations are techniques used to shift, stretch, compress, or reflect graphs on a coordinate grid. In your exercise, you shifted the graph by translating the coordinate axes to a new origin. This type of transformation is called a translation.When you perform graph transformations, you deal with modifications like these:

• **Translations**: Shifting the entire graph horizontally, vertically, or both. For example, replacing \(X\) with \((x+1)\) shifts the graph horizontally while replacing \(Y\) with \((y+2)\) shifts it vertically.
• **Reflections**: Flipping the graph over a specified axis, such as mirror-imaging the curve over the x-axis or y-axis.
• **Stretches and Compressions**: Expanding or compressing the graph along the x or y axes, altering the graph's width or height respectively.

Understanding these transformations enables students to predict and manipulate graph behaviors. As a result, pupils gain invaluable insights into the symmetry and shape shifts that graphs can undergo, thus broadening their problem-solving toolkits.