Problem 47 - Precalculus with Limits

Question

In Exercises 45-56, identify any intercepts and test for symmetry. Then sketch the graph of the equation. \( y = x^2 - 2x \)

Step-by-Step Solution

Answer

The intercepts of the equation \( y = x^2 - 2x \) are at (0,0) and (2,0). The graph is not symmetrical about the y-axis, x-axis or the origin. Finally, the graph is a parabola opening upward.

Step 1: Find the y-intercept

To find the y-intercept, we set \( x = 0 \) in the equation. This is because the y-intercept is the point where the graph intersects the y-axis, and the x-coordinate at this point is always zero. So, \( y = (0)^2 - 2*0 = 0 \). Therefore, the y-intercept is \( (0,0) \)

Step 2: Find the x-intercepts

To find the x-intercepts, we set \( y = 0 \) in the equation. This is because the x-intercept is the point where the graph intersects the x-axis, and the y-coordinate at this point is always zero. So, \( 0 = x^2 - 2x \). By factoring this equation, we get \( x(x - 2) = 0 \). This gives us two x-intercepts at \( (0,0) \) and \( (2,0) \)

Step 3: Test for symmetry

An equation is symmetric about the y-axis if replacing \( x \) with \( -x \) gives the same equation. Likewise, an equation is symmetric about the x-axis if replacing \( y \) with \( -y \) gives the same equation. And an equation is symmetric about the origin if replacing \( x \) with \( -x \) and \( y \) with \( -y \) gives the same equation. So, we replace \( x \) with \( -x \) in our equation to get \( y = (-x)^2 - 2*(-x) \), which simplifies to \( y = x^2 + 2x \). Since this is not the same as our initial equation, our graph is not symmetric about the y-axis. Similarly, there is no way to replace \( y \) with \( -y \) in our equation, so our graph is not symmetric about the x-axis. And because of the lack of symmetry in both the x-axis and y-axis, our graph is also not symmetric about the origin.

Step 4: Sketch the Graph

The given equation \( y = x^2 - 2x \) is a quadratic equation or a parabolic equation and we have already found points that the graph passes through: the intercepts. Using the fact that the coefficient of \( x^2 \) is positive, we can conclude the parabola will open upward. Now, juste draw the graph of this equation through the intercepts and shape it as an upwards opening parabola.

Key Concepts

Understanding Y-Intercepts

The y-intercept is a critical point on the graph where it crosses the y-axis. This point tells us the value of the function when the input, or x, is zero. To find the y-intercept, we substitute zero for x in the equation. For the equation \( y = x^2 - 2x \), when \( x = 0 \), substituting gives us \( y = (0)^2 - 2(0) = 0 \). Thus, the y-intercept is at point \((0, 0)\). This means at x = 0, the output, or y, is also 0.

Understanding the y-intercept helps in sketching the graph. It provides a starting point and can indicate symmetry or give clues about the overall shape of the graph. Always remember that the y-intercept is where the graph touches the y-axis, helping you begin your sketch.

Finding X-Intercepts

X-intercepts occur where the graph crosses the x-axis. At these points, the value of y is zero. To find x-intercepts, you set the equation to zero and solve for x. For \( y = x^2 - 2x \), you set \( y = 0 \), leading to the equation \( 0 = x^2 - 2x \).

By factoring, \( x(x - 2) = 0 \), we find the solutions \( x = 0 \) and \( x = 2 \). Hence, there are x-intercepts at \((0, 0)\) and \((2, 0)\). These intercepts tell us where the graph touches or crosses the x-axis. Plotting these points offers guidance on the extent and orientation of the graph.

• X-intercepts help define the breadth of the parabola.
• These points are essential to determine the roots of the quadratic equation.

A graph crossing or touching the x-axis at these points gives insight into its structure.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree. They generally have the form \( ax^2 + bx + c = 0 \), where a, b, and c are constants, and \( a eq 0 \). The equation \( y = x^2 - 2x \) is a simple quadratic equation with \( a = 1 \), \( b = -2 \), and \( c = 0 \).

Quadratics are known for their characteristic 'U' or 'n'-shaped curves called parabolas. An essential feature is that they have a vertex, which is the highest or lowest point of the parabola. For a quadratic equation like ours where the x-term dominates without any constant, symmetry is around the vertex.

The roots or solutions of a quadratic - the x-intercepts - give us a practical view of the curve's crossing points. Additionally, knowing the sign of the leading coefficient \( a \) tells us the parabola's direction:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

Understanding these properties simplifies sketching and analyzing quadratic functions.