Question

Consider f(x) = -(c+7)^2 + 4. Which of the following are true for f(x)? Check all that apply. Answer choices: the equation is a quadratic. The graph is linear. the vertex is (7,4). The axis of symmetry is x= -7. The y-intercept is (0,4). The graph has a relative maximum. The equation has no real solutions.​

Answer 1

The answers are presented below:

• The equation is a quadratic - TRUE
• The graph is linear. the vertex is (7,4) - FALSE
• The axis of symmetry is x= -7 - TRUE
• The y-intercept is (0,4)  - FALSE
• The graph has a relative maximum. - TRUE
• The equation has no real solutions.​ - FALSE

Quadratic function

The quadratic function can be represented by a quadratic equation in the Standard form: ax²+bx+c=0 where: a, b and c are your respective coefficients. In the quadratic function the coefficient "a" must be different than zero (a≠0) and the degree of the function must be equal to 2.

The vertex of an up-down facing parabola of the form ax²+bx+c is $x_v=\frac{-b}{2a}$  . Knowing the x-coordinate of vertex, you can find the y-coordinate of vertex. When a vertical line passes through the vertex of the parabola, it  is called the axis of symmetry.

The y-intercept is the point where the function crosses the y-axis. And, the x-intercept is the point where the function crosses the x-axis, in other words, when y=0.

First, you should convert the given equation to Standard form.

f(x) = -(c+7)² + 4

f(x)= - (c²+14c+49)+4

f(x)= -c²-14c-49+4

f(x)= -c²-14c-49+4

f(x)= -c²-14c-45

After that, you should check each option of the question.

• The equation is a quadratic

Correct because present a degree 2 -- f(x)= -c²-14c-45

• The graph is linear

Incorrect because it is a quadratic function and this function is represented by a parabola.

• The vertex is (7,4)

The coefficients are: a=-1, b=-14, c=-45. Therefore,

$c_v=\frac{-b}{2a}=-\frac{-14}{2*(-1)} =-\frac{-14}{-2}=-(7)=-7$

Incorrect because x-coordinate of vertex is -7

• The axis of symmetry is x= -7.

Correct because  $c_v=\frac{-b}{2a}=-\frac{-14}{2*(-1)} =-\frac{-14}{-2}=-(7)=-7$

• The y-intercept is (0,4)

You can find the y-intercept using c=0, then f(x)= -c²-14c-45 will be:

f(x)=-45.

Incorrect because the y-intercept is (0,-45).

• The graph has a relative maximum.

The coefficient a of a quadratic function is negative, thus the function has a point maximum. A relative maximum point is defined as the point where the function changes your direction. In other words, the values of the function increase before the relative maximum and the values decrease after the relative maximum.

Correct, see the attached image.

• The equation has no real solutions.​

It is possible to find the number of roots or solutions from discriminant.

For quadratic equation the discriminant is determined by D=b²-4ac, where: a, b and c are coefficients.

If D > 0 - the function will have two real solutions;

If D = 0 - the function will have one real solution;

If D < 0 - the function will have imaginary solutions;

For given equation, the coefficients are: a=-1, b=-14, c=-45. Thus, D=(-14)²-4*(-1)*(-45)

D=196-180=16

D=16 thus D > 0

Incorrect because the D>0.

Read more about the quadratic function here:

brainly.com/question/1497716