Sketch the parabola described by the equation 𝑦 = 2𝑥^2 − 20𝑥 + 42.
To sketch the parabola described by the equation 𝑦 = 2𝑥^2 − 20𝑥 + 42, we can first find the vertex. The x-coordinate of the vertex is given by 𝑥 = −𝑏/2𝑎, where 𝑏 is the coefficient of 𝑥 and 𝑎 is the coefficient of 𝑥^2. In this case, 𝑏 = −20 and 𝑎 = 2, so 𝑥 = −(−20)/(2×2) = 5.
Substituting 𝑥 = 5 into the equation, we find the y-coordinate of the vertex: 𝑦 = 2×5^2 − 20×5 + 42 = 2.
Therefore, the vertex of the parabola is (5, 2).
Next, we can find the y-intercept by setting 𝑥 = 0: 𝑦 = 2×0^2 − 20×0 + 42 = 42. The y-intercept is (0, 42).
To find the x-intercepts, we can solve the equation 𝑦 = 2𝑥^2 − 20𝑥 + 42 for 𝑥 when 𝑦 = 0:
0 = 2𝑥^2 − 20𝑥 + 42
Dividing both sides by 2:
0 = 𝑥^2 − 10𝑥 + 21
Factoring:
0 = (𝑥 − 3)(𝑥 − 7)
Therefore, the x-intercepts are (3, 0) and (7, 0).
We can now sketch the parabola using these points:
- The vertex is at (5, 2).
- The y-intercept is at (0, 42).
- The x-intercepts are at (3, 0) and (7, 0).
The parabola opens upward because the coefficient of 𝑥^2 is positive. The graph should look something like this:
To sketch the parabola described by the equation 𝑦 = 2𝑥^2 − 20𝑥 + 42, we can follow these steps:
1. Find the vertex of the parabola:
- The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a and b are coefficients in the equation 𝑦 = ax^2 + bx + c. In this case, a = 2 and b = -20.
- Substitute the values into the formula: x = -(-20) / (2*2) = 20 / 4 = 5.
- The y-coordinate of the vertex can be found by substituting the x-coordinate into the equation: 𝑦 = 2(5)^2 - 20(5) + 42 = 2(25) - 100 + 42 = 50 - 100 + 42 = -8.
- So, the vertex of the parabola is (5, -8).
2. Find the y-intercept:
- The y-intercept occurs when x = 0. Substitute x = 0 into the equation to find the y-coordinate: 𝑦 = 2(0)^2 - 20(0) + 42 = 0 - 0 + 42 = 42.
- The y-intercept is therefore (0, 42).
3. Find the x-intercepts:
- To find the x-intercepts, set y = 0 in the equation: 2x^2 - 20x + 42 = 0.
- Solve for x by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula here:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values a = 2, b = -20, and c = 42 into the formula:
x = (-(-20) ± √((-20)^2 - 4*2*42)) / (2*2)
x = (20 ± √(400 - 336)) / 4
x = (20 ± √64) / 4
x = (20 ± 8) / 4
x1 = (20 + 8) / 4 = 28 / 4 = 7
x2 = (20 - 8) / 4 = 12 / 4 = 3
- The x-intercepts are therefore (7, 0) and (3, 0).
4. Plot the vertex, y-intercept, and x-intercepts on a graph paper.
5. Draw a smooth curve passing through these points. The parabola opens upwards since the coefficient of the x^2 term (a) is positive.
The sketch of the parabola described by the equation 𝑦 = 2𝑥^2 − 20𝑥 + 42 should resemble a "U" shape with the vertex at (5, -8), passing through the points (0, 42), (7, 0), and (3, 0).