The axis of symmetry for a function in the form f(x) = x2 + 4x − 5 is x = −2. What are the coordinates of the vertex of the graph?

The quadratic function is described by the standard equation [tex]\boxed{ \ f(x) = ax^2 + bx + c \ }.[/tex] The graph of a quadratic function is called a parabola.

We have the quadratic function [tex]\boxed{ \ f(x) = x^2 + 4x - 5 \ }.[/tex]

We identify the coefficients a, b, and c. For this equation, [tex]\boxed{ \ a = 1, b = 4, and \ c = -5 \ }[/tex]
The axis of symmetry is [tex]\boxed{ \ x = h = -\frac{b}{2a} \ }[/tex], i.e., [tex]\boxed{ \ x = h = -\frac{4}{2(1)} \rightarrow h = -2 \ }.[/tex] Thus, the statement in the question for the symmetry axis x = -2 is correct.
The vertex (or turning point) is [tex]\boxed{ \ (h, k) \ },[/tex] where [tex]\boxed{ \ k = f(h) \ }[/tex] or [tex]\boxed{ \ k = \frac{b^2 - 4ac}{-4a} \ }[/tex]
The parabola opens upward because a > 0, resulting in a vertex that is a minimum.

Finding the minimum value is as follows:

[tex]\boxed{ \ k = f (-2) = (-2)^2 + 4(-2) - 5 \rightarrow k = -9 \ }, or[/tex]
[tex]\boxed{ \ k = \frac{4^2 - 4(1)(-5)}{-4(1)} = -\frac{36}{4} \rightarrow k = -9 \ }[/tex]

Therefore, the coordinates of the vertex of the graph are (h, k), i.e., [tex]\boxed{\boxed{ \ (-2, -9) \ }}[/tex]

Other components:

The y-intercept of the quadratic function f(x) = x² + 4x - 5 is (0, c), i.e., the point [tex]\boxed{ \ (0, -5) \ }.[/tex]
The factorization of f (x) = x² + 4x - 5 becomes f(x) = (x + 5)(x - 1). Then we get the x-intercepts at [tex]\boxed{ \ (-5, 0) \ and \ (1, 0) \ }[/tex]

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Keywords: the axis of symmetry, for a function, f(x) = x² + 4x − 5, x = −2, which is the graph of f(x) = (x - 1)(x + 4), what, the coordinates of the vertex of the graph, the x-intercept, quadratic function, a standard equation, the y-intercept, parabola, upward, the minimum value

Lanuel Lanuel · Answer 2 · 2022-06-03T22:08:43+02:00

Based on the calculations, the coordinates of the vertex (h, k) of this graph are equal to (-2, -9).

The standard form of a quadratic equation.

In Mathematics, the standard form of a quadratic equation is given by;

ax² + bx + c = 0

Note: The graph of any quadratic function is known as a parabola.

From the given quadratic function, we have:

f(x) = x² + 4x - 5

For its coefficients, we have:

a = 1.

b = 4.

c = -5.

Next, we would determine the coordinates of the vertex of this graph:

x = h = -b/2a

h = -4/2(1)

h = -4/2

h = -2.

Also, we can deduce that the parabola of this graph opens upward because "a" is greater than zero (0), which produces a vertex that is a minimum:

f(h) = k

f(-2) = k = (-2)² + 4(-2) - 5

f(-2) = k = 4 - 8 - 5

k = -9.

Vertex (h, k) = (-2, -9).

Read more on vertex here: brainly.com/question/525947

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