consider f(x) = -(x+7)^2 +4. check all that apply -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

the equation is a quadratic ✓ The function f(x) = -(x + 7)^2 + 4 is a quadratic equation because it can be written in the general form ax^2 + bx + c, where a, b, and c are constants. In this case, we have a negative coefficient for the squared term, which indicates that the graph will open downwards.

- the graph is linear ✗ The graph of a quadratic function is not linear; it is a parabola. A linear graph is represented by a straight line, which is not the case for a quadratic function.

- the vertex is (7,4) ✗ The vertex of the parabola is found at the point (-h, k) when the quadratic is in the vertex form f(x) = a(x - h)^2 + k. Since we have f(x) = -(x + 7)^2 + 4, the vertex is actually at (-7, 4), not (7, 4).

- the axis of symmetry is x=-7 ✓ The axis of symmetry for a parabola in vertex form f(x) = a(x - h)^2 + k is x = h. Since h is -7 in this case, the axis of symmetry is x = -7.

- the y-intercept is (0,4) ✗ To find the y-intercept, we set x to 0. If we plug x = 0 into the equation, we get f(0) = -(0 + 7)^2 + 4 = -49 + 4 = -45. Therefore, the y-intercept is (0, -45), not (0, 4).

- the graph has a relative maximum ✓ Since the coefficient of the x^2 term is negative (a = -1), the parabola opens downwards, and therefore, the vertex represents a relative maximum point on the graph.

- the equation has no real solutions ✗ This statement is incorrect. A real solution of a quadratic equation corresponds to the points where the parabola intersects the x-axis. Even if this particular parabola does not cross the x-axis and therefore has no real x-intercepts (which can be the case if the discriminant b^2 - 4ac is negative), the quadratic equation itself always has real (or complex) roots. In this case, since the function does not have an x-intercept, it has no real solutions; however, it still has two complex solutions.

Extra: - Quadratic Functions: These are functions that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and "a" is not equal to zero. The graph of a quadratic function is a curve called a parabola that can either open upwards (if a > 0) or downwards (if a < 0).

- Vertex of a Parabola: The highest or lowest point on a parabola is called the vertex. If the parabola opens upward, the vertex is the minimum point. If it opens downward, as in this case, the vertex is the maximum point.