Given a quadratic equation in the form y = ax2 +bx +c , and four choices of graphs representing that equation, how can you tell which choices could be eliminated right away by looking at the equation and looking at the graphs and doing some minimal calculations and not actually plotting any of the points? Please be specific and provide an example of an equation.
Wouldn't it be great to see the four choices?
At least, from the sign of a, one can tell whether it is concave upward or downward
a>0 => concave upward.
To determine which choices of graphs could be eliminated without plotting any points, you can look for certain characteristics in the equation and compare them to the characteristics of the graphs. Here are some steps to follow:
1. Find the discriminant: The discriminant is given by the formula b^2 - 4ac in the quadratic equation. The discriminant provides information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a double root). And if it is negative, the equation has no real roots (two complex roots).
2. Look for symmetry: Quadratic equations have a line of symmetry that passes through the vertex of the parabola. The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (h, k), where h = -b/(2a) and k = f(h). Identifying the axis of symmetry can help narrow down the choices.
3. Analyze the coefficient of x^2: The coefficient of x^2 (a) determines the steepness of the parabola. A positive value of a leads to a parabola opening upwards, while a negative value leads to a parabola opening downwards.
Example:
Let's consider the following quadratic equation: y = 2x^2 - 3x + 1
1. Discriminant: The discriminant is given by b^2 - 4ac = (-3)^2 - 4(2)(1) = 9 - 8 = 1. Since the discriminant is positive, this equation has two distinct real roots.
2. Symmetry: The axis of symmetry is given by x = -b/(2a) = -(-3)/(2 * 2) = 3/4. This means the parabola is symmetric about x = 3/4.
3. Coefficient of x^2: The coefficient of x^2 is 2, indicating that the parabola opens upward.
Now, let's consider the four choices of graphs. By analyzing the characteristics derived from the equation, we can eliminate certain choices:
- If a graph has a parabola that opens downward, it can be eliminated since the equation has a positive coefficient for x^2.
- If a graph does not have a line of symmetry at x = 3/4, it can be eliminated as it does not match the equation.
- If a graph does not have two distinct real roots, it can be eliminated.
By following these steps, you can narrow down the choices of graphs without plotting any points.