Solve systems by graphing linear and quadratic equations. Grade 11 Math 20-1.
Lesson 2.2 of Linear Functions & Systems of Equations in Math 20-1: Alberta curriculum lessons.
What is a Linear-Quadratic System?. A linear-quadratic system is a pair of equations where one is a line and the other is a parabola. When you graph both on the same grid, the points where they intersect are the solutions.
There are three possible scenarios depending on how the line meets the parabola:
What is a Tangent Line?. A tangent line touches a parabola at exactly one point without crossing it. This corresponds to the one-solution case for a linear-quadratic system.
What is a Quadratic-Quadratic System?. A quadratic-quadratic system is a pair of equations where both are parabolas. There are four possible scenarios:
Infinitely Many Solutions. Infinite solutions occur when both equations describe the same parabola. They are identical, just written differently. Every point on the parabola satisfies both equations.
Follow these steps whenever you solve a system of equations using a graphing calculator:
Key Reminder: Always Verify. After finding an intersection point, always substitute the coordinates back into both original equations to confirm it satisfies them. A single verification error can cost you the question.
Setup: y + 3x^2 - 2x - 4 = 0 and y + x + 2 = 0. Step 1: Isolate y in both equations.
From equation 1:
y = -3x^2 + 2x + 4
From equation 2:
y = -x - 2
Step 2 and 3: Find and Verify Intersections. Step 2: Graph both on the calculator. The two graphs intersect at (-1, -1) and (2, -4).
Step 3: Verify (-1, -1).
Equation 1: -1 + 3(-1)^2 - 2(-1) - 4 = -1 + 3 + 2 - 4 = 0 ✓
Equation 2: -1 + (-1) + 2 = 0 ✓
Verify (2, -4).
Equation 1: -4 + 3(2)^2 - 2(2) - 4 = -4 + 12 - 4 - 4 = 0 ✓
Equation 2: -4 + 2 + 2 = 0 ✓
Solutions: (-1, -1) and (2, -4)
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