Respuesta :
Using the discriminant of a quadratic equation, if the graph is translated shifted up 4 units, the graph will have no x-intercepts.
What is the discriminant of a quadratic equation and how does it influence the solutions?
A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The discriminant is:
[tex]\Delta = b^2 - 4ac[/tex]
The solutions are as follows:
- If [tex]\mathbf{\Delta > 0}[/tex], and it has 2 x-intercepts.
- If [tex]\mathbf{\Delta = 0}[/tex], it has 1 x-intercept.
- If [tex]\mathbf{\Delta < 0}[/tex], it has no x-intercepts.
In this problem, the function is given by:
f(x) = (x + 5)² - 3.
In standard form:
f(x) = x² + 10x + 22.
We want to find coefficient k for which the function has [tex]\Delta < 0[/tex], then:
f(x) = x² + 10x + 22 + k.
The coefficients are a = 1, b = 10, c = 22 + k, hence:
[tex]\Delta < 0[/tex]
10² - 4(22 + k) < 0
100 - 88 - 4k < 0
4k > 12
k > 3.
Hence, with k = 4, the function is shifted up 4 units, and the graph will have no x-intercepts.
More can be learned about the discriminant of a quadratic equation at https://brainly.com/question/19776811
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