Problem 19 - Precalculus with Limits

Question

In Exercises 19-22, graphically estimate the \( x \)- and \( y \)-intercepts of the graph. Verify your results algebraically. \( y = (x-3)^2 \)

Step-by-Step Solution

Answer

The x-intercept of the function \( y = (x-3)^2 \) is 3 and the y-intercept is 9. This solution was both graphically estimated and algebraically verified.

Step 1: Estimate graphically

Begin by sketching a graph of the function \( y = (x-3)^2 \). The graph is a parabola that opens upwards with its vertex at the point (3,0). From the graph, it may be estimated that the x-intercept is 3 and the y-intercept is 9.

Step 2: Verify algebraically: x-intercept

To find the x-intercept, the equation \( y = (x-3)^2 \) is set to be equal to zero:0 = (x-3)^2. Solving this, we find x-3 = 0, which implies x = 3. Thus, the x-intercept of the function is 3 as estimated from the graph.

Step 3: Verify algebraically: y-intercept

To find the y-intercept of the function, the value of y is determined when x=0. Substituting x = 0 into the equation \( y = (x-3)^2 \) gives:y = (0-3)^2 = 9.Hence, the y-intercept is 9. This confirms the graphical estimate.

Key Concepts

Graphing Functions

Graphing functions is a foundational math skill that helps visualize equations. When graphing a function, like the parabola described by \( y = (x-3)^2 \), plot enough points to capture its shape. For parabolas, identify key features such as:

• The vertex: the top or bottom point of the parabola. In this case, it's (3,0).
• The axis of symmetry: a vertical line through the vertex, guiding the parabola's balance. Here, it's \( x = 3 \).
• Intercepts: points where the graph crosses the axes. Estimate these carefully on your sketched graph.

Graphing functions helps visually identify these elements and provides an effective way to understand the behavior of algebraic expressions.

Algebraic Verification

Algebraic verification is a powerful method to confirm graphical estimates. This involves using algebraic techniques to validate features like intercepts. For instance:

• Finding x-intercepts involves setting \( y = 0 \). For \( y = (x-3)^2 \), solving \( 0 = (x-3)^2 \) confirms the x-intercept is \( x = 3 \).
• Finding y-intercepts requires setting \( x = 0 \). For our function, substituting \( x = 0 \) yields \( y = 9 \), confirming the y-intercept.

Algebraic verification ensures the reliability of graphical interpretations by providing precise calculations, thus reinforcing the understanding of mathematical concepts.

Parabolas

Parabolas are U-shaped graphs representing quadratic functions, common in algebra and graphing exercises. The standard form \( y = ax^2 + bx + c \) highlights parabolas' distinctive features:

• A vertex that represents the parabola's highest or lowest point.
• An axis of symmetry, which helps in drawing the parabola mirror-like around it.
• Intercepts, where the parabola meets the x-axis and y-axis. These points are crucial for fully understanding the graph's position.

For \( y = (x-3)^2 \), the parabola is vertex form, showing vertex at (3,0). Understanding parabolas involves spotting these characteristics to fully grasp the behavior and properties of quadratic functions.