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Answer
The graphs intersect at (0, –3) and (1, –4).
Solution
We are given the line
y = –x – 3
and the quadratic
y = 2x² – 3x – 3.
To find their points of intersection, we set the two expressions for y equal to each other:
–x – 3 = 2x² – 3x – 3.
Step 1. Simplify the equation by adding x + 3 to both sides:
–x – 3 + x + 3 = 2x² – 3x – 3 + x + 3
0 = 2x² – 2x.
Step 2. Factor the right-hand side:
2x² – 2x = 2x(x – 1) = 0.
Set each factor equal to 0:
2x = 0 → x = 0,
x – 1 = 0 → x = 1.
Step 3. Find the corresponding y-values using the line equation y = –x – 3:
• When x = 0:
y = –(0) – 3 = –3,
so one intersection point is (0, –3).
• When x = 1:
y = –(1) – 3 = –4,
so the other intersection point is (1, –4).
Thus, the graphs of y = –x – 3 and y = 2x² – 3x – 3 intersect at (0, –3) and (1, –4).
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Simplify this solution Beyond the Answer
These two equations represent a linear function and a quadratic function, which intersect at points where their outputs, or \(y\)-values, are equal. To find these intersection points, you would typically set them equal to each other: \(-x - 3 = 2x^2 - 3x - 3\). This gives you a quadratic equation to solve! Next, remember that graphing both functions can provide a visual sense of their behavior and how many intersections they have. It’s exciting to see the straight line cut through the parabola at specific points, revealing solutions that are both tangible and visually engaging. Keep your graphing tools handy!
