B. Draw the graphs of the following functions. 1. y=x² 2. y=(x-2)² 3. y=(x+2)² 4. y=(x+1)' 5. y=(x-1)' a. Analyze the graphs. b. What do you notice about the graphs of quadratic functions whose equations are of the form y=(x-h)²? c. How would you compare the graph of y=(x-h) and that of y=x? d. Discuss your ideas and observations. give me step by step
Answer (1)
Answer: "Junior High School" level. The question asks toa. Draw the graphs of several functions:1. y = x^22. y = (x-2)^23. y = (x+2)^24. y = (x+1)^25. y = (x-1)^2b. Analyze the graphs.c. Discuss what is noticed about the graphs of quadratic functions whose equations are of the form y = (x-h)^2.d. Compare the graph of y = (x-h)^2 and that of y = x^2.e. Discuss ideas and observations.f. Provide a step-by-step explanation.Here's a step-by-step approach to answer the question:Step 1: Understand the Parent Function y = x^2The function y = x^2 is the most basic quadratic function, often called the parent function for parabolas that open upwards. Its vertex is at the origin (0,0).To graph it, pick some x-values and calculate the corresponding y-values: * If x = -2, y = (-2)^2 = 4 * If x = -1, y = (-1)^2 = 1 * If x = 0, y = (0)^2 = 0 * If x = 1, y = (1)^2 = 1 * If x = 2, y = (2)^2 = 4Plot these points and draw a smooth U-shaped curve.Step 2: Graph Functions of the Form y = (x-h)^2The general form y = (x-h)^2 represents a horizontal translation of the parent function y = x^2. * For y = (x-2)^2: Here, h = 2. This means the graph of y = x^2 is shifted 2 units to the right. The vertex will be at (2,0). * If x = 0, y = (0-2)^2 = 4 * If x = 1, y = (1-2)^2 = 1 * If x = 2, y = (2-2)^2 = 0 * If x = 3, y = (3-2)^2 = 1 * If x = 4, y = (4-2)^2 = 4 * For y = (x+2)^2: Here, h = -2 (because x+2 can be written as x-(-2)). This means the graph of y = x^2 is shifted 2 units to the left. The vertex will be at (-2,0). * If x = -4, y = (-4+2)^2 = 4 * If x = -3, y = (-3+2)^2 = 1 * If x = -2, y = (-2+2)^2 = 0 * If x = -1, y = (-1+2)^2 = 1 * If x = 0, y = (0+2)^2 = 4 * For y = (x+1)^2: Here, h = -1. Shift 1 unit to the left. Vertex at (-1,0). * If x = -3, y = (-3+1)^2 = 4 * If x = -2, y = (-2+1)^2 = 1 * If x = -1, y = (-1+1)^2 = 0 * If x = 0, y = (0+1)^2 = 1 * If x = 1, y = (1+1)^2 = 4 * For y = (x-1)^2: Here, h = 1. Shift 1 unit to the right. Vertex at (1,0). * If x = -1, y = (-1-1)^2 = 4 * If x = 0, y = (0-1)^2 = 1 * If x = 1, y = (1-1)^2 = 0 * If x = 2, y = (2-1)^2 = 1 * If x = 3, y = (3-1)^2 = 4(a. Draw the graphs): On a coordinate plane, draw all five parabolas. Use different colors or labels to distinguish them.Step 3: Analyze the Graphs (b. and c.) * Shape: All the graphs maintain the same parabolic shape as y = x^2. They are congruent (same size and shape). * Orientation: All parabolas open upwards. * Vertex Location: * For y = x^2, the vertex is at (0,0). * For y = (x-2)^2, the vertex is at (2,0). * For y = (x+2)^2, the vertex is at (-2,0). * For y = (x+1)^2, the vertex is at (-1,0). * For y = (x-1)^2, the vertex is at (1,0). * Axis of Symmetry: The axis of symmetry for y = x^2 is the y-axis (x=0). For y = (x-h)^2, the axis of symmetry is the vertical line x = h.Step 4: Compare y = (x-h)^2 and y = x^2 (d.) * The graph of y = (x-h)^2 is a horizontal translation of the graph of y = x^2. * If h is positive (e.g., y = (x-2)^2, where h=2), the graph of y = x^2 is shifted h units to the right. The vertex moves from (0,0) to (h,0). * If h is negative (e.g., y = (x+2)^2, which is y = (x-(-2))^2, where h=-2), the graph of y = x^2 is shifted |h| units to the left. The vertex moves from (0,0) to (h,0). * The shape and size of the parabola remain identical; only its position on the x-axis changes.Step 5: Discuss Ideas and Observations (e.) * The parameter 'h' in the equation y = (x-h)^2 directly controls the horizontal shift of the parabola. * A "minus h" inside the parentheses (e.g., x-2) results in a shift to the right (positive x-direction). * A "plus h" inside the parentheses (e.g., x+2, which is x-(-2)) results in a shift to the left (negative x-direction). * This form of the quadratic equation, y = (x-h)^2, is a simplified version of the vertex form y = a(x-h)^2 + k, where a=1 and k=0. In this case, the vertex is always on the x-axis.This comprehensive explanation covers all parts of the user's request.
