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?
How can we answer without looking at the four choices of graphs?
See the following in regards to graphing quadratic functions.
http://www.purplemath.com/modules/grphquad2.htm
The graph of this equation will be which conic section?(x-5)^2+y^2=64
To determine which choices can be eliminated right away without plotting any points, you can follow these steps:
1. Look at the coefficient of the quadratic term (a). If it is positive, the graph will open upward, and if it is negative, the graph will open downward.
2. Examine the discriminant (b^2 - 4ac) of the quadratic equation. The discriminant tells us the number of solutions the equation has.
a. If the discriminant is positive, the equation will have two distinct real solutions. This means the graph will intersect the x-axis at two different points.
b. If the discriminant is zero, the equation will have one real solution (a double root). This means the graph will touch the x-axis at one point.
c. If the discriminant is negative, the equation will have no real solutions. This means the graph will not intersect the x-axis and will stay entirely above or below it.
Using these observations, you can analyze the provided choices and eliminate the ones that contradict these characteristics.
To determine which choices of graphs can be eliminated without plotting any points, you need to analyze the quadratic equation and consider the properties of quadratic functions.
Here's a step-by-step approach:
1. Consider the leading coefficient (a):
- If a > 0, the graph will open upwards (U-shape) because the positive coefficient amplifies the quadratic term (x^2). The vertex will be the lowest point on the graph.
- If a < 0, the graph will open downwards (narrow U-shape) because the negative coefficient flips the quadratic term (x^2). The vertex will be the highest point on the graph.
2. Evaluate the discriminant (b^2 - 4ac):
- If the discriminant is positive (b^2 - 4ac > 0), the quadratic equation has two distinct real roots. The graph will intersect the x-axis at two points.
- If the discriminant is zero (b^2 - 4ac = 0), the quadratic equation has one real root. The graph will touch the x-axis at one point (tangent).
- If the discriminant is negative (b^2 - 4ac < 0), the quadratic equation has no real roots. The graph will not intersect the x-axis.
3. Examine the constant term (c):
- If c > 0, the graph will have a y-intercept above the x-axis.
- If c < 0, the graph will have a y-intercept below the x-axis.
By analyzing the equation and utilizing these properties, you can eliminate the choices that do not match the given information. For example:
- If the equation has a positive leading coefficient (a > 0) and a positive constant term (c > 0), eliminate any graphs that open downwards (a < 0) or have a y-intercept below the x-axis (c < 0).
- If the discriminant is positive (two real roots), eliminate any graphs that do not intersect the x-axis at two distinct points.
- If the discriminant is zero (one real root), eliminate any graphs that do not touch the x-axis at one point.
By eliminating the options that contradict the properties of the equation, you can narrow down the choices to find the correct graph representation without plotting any points.