To predict the number of x-intercepts of a quadratic function without drawing the graph or fully solving the equation, you can use the discriminant, which is a part of the quadratic formula. The discriminant tells you the nature of the roots (x-intercepts) of the quadratic equation without actually solving for them.
Given a quadratic function in the form f(x) = ax^2 + bx + c, the discriminant can be calculated using the formula:
Discriminant (D) = b^2 - 4ac
Now, based on the value of the discriminant, we can determine the number of x-intercepts:
1. D > 0 (Positive discriminant): If the discriminant is positive, it means that b^2 - 4ac is greater than 0. In this case, the quadratic equation has two distinct real roots (x-intercepts). The graph of the quadratic function will intersect the x-axis in two different points.
2. D = 0 (Zero discriminant): If the discriminant is zero, it means that b^2 - 4ac equals 0. In this case, the quadratic equation has exactly one real root (x-intercept). The graph of the quadratic function will touch the x-axis at a single point.
3. D < 0 (Negative discriminant): If the discriminant is negative, it means that b^2 - 4ac is less than 0. In this case, the quadratic equation has no real roots (x-intercepts). The graph of the quadratic function does not intersect the x-axis.
Now, moving on to the next part of your question:
If the graph of f(x) = ax^2 + bx + c has x-intercepts (m,0) and (n,0), we can determine the x-intercepts of the function g(x) = –ax^2 – bx – c by finding the roots of g(x) = 0.
To find the roots of a quadratic equation, we need to set g(x) equal to zero and solve for x:
–ax^2 – bx – c = 0
We can multiply through by -1 to make it easier to work with:
ax^2 + bx + c = 0
Now, we can observe that the quadratic equation for g(x) is the same as the quadratic equation for f(x), except with opposite signs for all coefficients: a, b, and c.
Therefore, the x-intercepts of g(x) are also (m,0) and (n,0), but with a negative y-value since the signs are opposite. This means that the x-intercepts of g(x) are (-m,0) and (-n,0).
So, if f(x) has x-intercepts (m,0) and (n,0), then g(x) will have x-intercepts (-m,0) and (-n,0).
I hope this explanation helps! Let me know if you have any further questions.