- For problem ①: The line y = -2x - 6 has a y-intercept at (0, -6) and a slope of -2. The parabola y = x² - 6x + 3 opens upwards with vertex at (3, -6). They intersect at approximately (1.3, -8.6) and (4.7, -15.4).
- For problem ②: The horizontal line y = -4 intersects the upward-opening parabola y = x² - 2x - 3 at points where x² - 2x - 3 = -4, which simplifies to x² - 2x + 1 = 0, giving a single point of tangency at (1, -4).
- For problem ③: The line y = x - 5 has a y-intercept at (0, -5) and slope 1. The downward-opening parabola y = -x² + x - 5 has vertex at (0.5, -4.75). They intersect at (0, -5) and (1, -4).
- For problem ④: The line y = -4x + 25/2 has a y-intercept at (0, 12.5) and slope -4. The upward-opening parabola y = (1/2)x² + 4x + 4 has vertex at (-4, -4). They intersect at approximately (-9.1, 48.9) and (1.1, 20.6).
- For problem ⑤: The line y = x - 1 has a y-intercept at (0, -1) and slope 1. The upward-opening parabola y = x² - 2x + 4 has vertex at (1, 3). They do not intersect; the parabola lies entirely above the line.
- For problem ⑥: The line y = -2x - 6 has a y-intercept at (0, -6) and slope -2. The upward-opening parabola y = x² - 6x + 3 has vertex at (3, -6). They intersect at approximately (0.6, -7.2) and (5.4, -16.8).
Parent Tip: Review the logic above to help your child master the concept of systems of linear and quadratic equations worksheet.
