Can you explain this?
An intercept of the quadratic graph is the same as the root of quadratic equation. You have to find out the discriminant. If the discriminant is positive, you have two roots which equals two intercepts. If it is zero, there is one root which equals one intercept. If it is negative, there are no roots and no intercepts.
The graph of f(x)=-ax^2-bx-c can have no x-intercepts because all integers are negative.
not so. Just pick a,b,c so that b^2-4ac is positive.
f(x) = -x^2-10x-3
b^2 is positive whether b is positive or negative.
It was all good until the last line. That's just rubbish.
To explain the relationship between intercepts of a quadratic graph and roots of a quadratic equation, we need to understand how to find the intercepts and roots.
1. Intercept:
In the context of a quadratic graph, an intercept refers to the points where the graph intersects the x-axis. These points are where the value of y (or f(x)) is equal to zero.
2. Roots of a quadratic equation:
A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The roots of the equation are the values of x that make the equation true, i.e., when the left side (ax^2 + bx + c) equals zero.
Now, let's discuss how to relate the intercepts and roots based on the discriminant of the quadratic equation.
3. Discriminant:
The discriminant (denoted as Δ) is a value calculated from the coefficients a, b, and c of a quadratic equation. It helps determine the nature and number of roots the equation has.
The formula to calculate the discriminant is:
Δ = b^2 - 4ac
Now, let's consider the three cases based on the value of the discriminant:
a. If Δ > 0:
In this case, the discriminant is positive, indicating that there are two distinct roots. As a result, the quadratic graph will intersect the x-axis at two different points, which are the two intercepts.
b. If Δ = 0:
Here, the discriminant is zero, meaning there is only one repeated root. Consequently, the quadratic graph will touch the x-axis at a single point, which represents the intercept of the graph.
c. If Δ < 0:
When the discriminant is negative, there are no real roots for the quadratic equation. As a result, the graph will not intersect the x-axis at any point, indicating no intercepts.
So, if all the coefficients in the quadratic equation (such as a, b, c) are negative integers, as mentioned in your example, it is possible for the quadratic graph to not have any x-intercepts since the equation may have no real roots.