If a graph of a quadratic function can have 0, 1 or 2 x-intercepts. How can you predict the number of x-intercepts without drawing the graph or (completely) solving the related equation?
Suppose that the graph of f(x) = ax^2+bx+c has x-intercepts (m,0) and (n,0). What are the x-intercepts of g(x) = –ax^2–bx–c?
I really don't understand this, any elaboration will be greatly appreciated. thank in advance
look at
b^2 - 4 a c
if +, then two intercepts
if 0, then just grazes the x axis, one intercept
if -, then never hits the axis, imaginary roots only
if a x^2 + b x + c = 0
then
- (a x^2 + b x + c) = 0
is the same because +0 = -0
To predict the number of x-intercepts without drawing the graph or solving the equation, you can look at the discriminant of the quadratic equation. The discriminant is the part inside the square root of the quadratic formula, which is given by Δ = b^2 - 4ac.
If Δ > 0, there are two distinct x-intercepts.
If Δ = 0, there is one x-intercept (with a multiplicity of 2).
If Δ < 0, there are no x-intercepts.
For the quadratic function f(x) = ax^2 + bx + c, you are given that it has x-intercepts (m,0) and (n,0). This means that f(m) = 0 and f(n) = 0.
Now, consider the quadratic function g(x) = -ax^2 - bx - c. To find its x-intercepts, you need to solve the equation g(x) = 0.
Substituting the value of g(x) into the equation, we get:
-ax^2 - bx - c = 0
Dividing by -1 (to simplify the equation), we have:
ax^2 + bx + c = 0
Comparing this equation to the original equation f(x) = ax^2 + bx + c, we can see that they have the same coefficients a, b, and c. Therefore, the x-intercepts of g(x) are the same as the x-intercepts of f(x), namely (m,0) and (n,0).
To predict the number of x-intercepts of a quadratic function without drawing the graph or completely solving the related equation, you can look at the discriminant (Δ) of the quadratic equation.
The discriminant is the term inside the square root in the quadratic formula, Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
By analyzing the value of the discriminant, you can determine how many x-intercepts the quadratic function has:
1. If the discriminant Δ > 0, then the quadratic function has two distinct x-intercepts. This means the graph will intersect the x-axis at two different points.
2. If the discriminant Δ = 0, then the quadratic function has exactly one x-intercept. This means the graph will touch the x-axis at a single point.
3. If the discriminant Δ < 0, then the quadratic function has no real x-intercepts. This means the graph does not intersect the x-axis and remains above or below it.
Using this information, you can predict the number of x-intercepts of a quadratic function without drawing the graph or solving the equation.
Now, let's move on to the second part of your question.
Given the quadratic function f(x) = ax^2 + bx + c with x-intercepts (m, 0) and (n, 0), we can find the x-intercepts of the function g(x) = -ax^2 - bx - c.
To find the x-intercepts of g(x), we need to solve the equation g(x) = 0.
Substituting g(x) = -ax^2 - bx - c, we get:
-ax^2 - bx - c = 0
Rearranging the equation to have zero on one side, we have:
ax^2 + bx + c = 0
This equation is similar to the one given for f(x), but with the opposite sign.
Since we know that x-intercepts of f(x) are (m, 0) and (n, 0), we can substitute these values into the equation:
a(m)^2 + b(m) + c = 0
a(n)^2 + b(n) + c = 0
By solving this system of equations, you can find the x-intercepts of g(x) = -ax^2 - bx - c.