The 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?

- a x ^ 2 - b x - c = - ( a x ^ 2 + b x + c )

Quadratic function can be concave up or concave down.

If a x ^ 2 + b x + c concave up or concave down graph of - a x ^ 2 - b x - c will be concave down or concave up.

The graph is rotated 180 ° around the x - axis, but x-intercepts stay same.

So x-intercepts of - a x ^ 2 - b x - c = x-intercepts of a x ^ 2 + b x + c

( m , 0 ) , ( n , 0 )

How can you predict the number of x-intercepts without drawing the graph.

Discriminant = b ^ 2 - 4 a c

If the discriminant is positive, then there are two distinct roots ( x - intercepts ).

If the discriminant is zero, then there is exactly one real root x - intercept ).

If the discriminant is negative, then there are no real roots( no one x - intercepts ).

x intercepts -3,1 points on the graph (

2,2.5)

Which of the following illustration a quadratic function?

|. g(x)=2 - x²
||. f(X) =6x-4+3×3
|||. y=2ײ -3 x +5
IV. C(x)=x(x-2)+2r²

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.