Answer :
Let's go through each part of the question thoroughly:
(a) Vertex of [tex]\( f \)[/tex]:
The function given is [tex]\( f(x) = x^2 + 4x - 21 \)[/tex]. To find the vertex of a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], we use the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -21 \)[/tex]. Plugging in the values:
[tex]\[ x = -\frac{4}{2 \times 1} = -2 \][/tex]
Substituting [tex]\( x = -2 \)[/tex] back into the function to find the y-coordinate of the vertex:
[tex]\[ f(-2) = (-2)^2 + 4(-2) - 21 = 4 - 8 - 21 = -25 \][/tex]
So, the vertex is [tex]\((-2, -25)\)[/tex].
(b) x-intercepts of the graph of [tex]\( f \)[/tex]:
The x-intercepts are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. We solve the equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex].
The solutions to this can be found by factoring or using the quadratic formula. The factors of [tex]\(-21\)[/tex] that add up to [tex]\(4\)[/tex] are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex]. Therefore, the factored form is:
[tex]\[ (x - 3)(x + 7) = 0 \][/tex]
Setting each factor to zero gives the solutions:
[tex]\( x - 3 = 0 \)[/tex] which gives [tex]\( x = 3 \)[/tex]
[tex]\( x + 7 = 0 \)[/tex] which gives [tex]\( x = -7 \)[/tex]
Thus, the x-intercepts are [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex].
(c) Solve [tex]\( f(x) = -21 \)[/tex] for [tex]\( x \)[/tex]:
We set the equation [tex]\( f(x) = -21 \)[/tex] equal to [tex]\( x^2 + 4x - 21 = -21 \)[/tex].
Simplifying that, we have:
[tex]\[ x^2 + 4x - 21 = -21 \][/tex]
Adding 21 to both sides:
[tex]\[ x^2 + 4x = 0 \][/tex]
Factoring out the common factor:
[tex]\[ x(x + 4) = 0 \][/tex]
Setting each factor to zero gives the solutions:
[tex]\( x = 0 \)[/tex] and [tex]\( x + 4 = 0 \)[/tex] which gives [tex]\( x = -4 \)[/tex]
These solutions indicate the points [tex]\((0, -21)\)[/tex] and [tex]\((-4, -21)\)[/tex] are on the graph.
(d) Graphing [tex]\( f(x) = x^2 + 4x - 21 \)[/tex]:
Using the information:
- Vertex: [tex]\((-2, -25)\)[/tex]
- x-intercepts: [tex]\((-7, 0)\)[/tex] and [tex]\((3, 0)\)[/tex]
- Points from [tex]\( f(x) = -21 \)[/tex]: [tex]\((0, -21)\)[/tex] and [tex]\((-4, -21)\)[/tex]
You can plot these points on a coordinate plane to sketch the graph of the function. The parabola will open upwards, as the coefficient of [tex]\( x^2 \)[/tex] is positive. These critical points help define the curve's shape and location accurately.
(a) Vertex of [tex]\( f \)[/tex]:
The function given is [tex]\( f(x) = x^2 + 4x - 21 \)[/tex]. To find the vertex of a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], we use the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -21 \)[/tex]. Plugging in the values:
[tex]\[ x = -\frac{4}{2 \times 1} = -2 \][/tex]
Substituting [tex]\( x = -2 \)[/tex] back into the function to find the y-coordinate of the vertex:
[tex]\[ f(-2) = (-2)^2 + 4(-2) - 21 = 4 - 8 - 21 = -25 \][/tex]
So, the vertex is [tex]\((-2, -25)\)[/tex].
(b) x-intercepts of the graph of [tex]\( f \)[/tex]:
The x-intercepts are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. We solve the equation [tex]\( x^2 + 4x - 21 = 0 \)[/tex].
The solutions to this can be found by factoring or using the quadratic formula. The factors of [tex]\(-21\)[/tex] that add up to [tex]\(4\)[/tex] are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex]. Therefore, the factored form is:
[tex]\[ (x - 3)(x + 7) = 0 \][/tex]
Setting each factor to zero gives the solutions:
[tex]\( x - 3 = 0 \)[/tex] which gives [tex]\( x = 3 \)[/tex]
[tex]\( x + 7 = 0 \)[/tex] which gives [tex]\( x = -7 \)[/tex]
Thus, the x-intercepts are [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex].
(c) Solve [tex]\( f(x) = -21 \)[/tex] for [tex]\( x \)[/tex]:
We set the equation [tex]\( f(x) = -21 \)[/tex] equal to [tex]\( x^2 + 4x - 21 = -21 \)[/tex].
Simplifying that, we have:
[tex]\[ x^2 + 4x - 21 = -21 \][/tex]
Adding 21 to both sides:
[tex]\[ x^2 + 4x = 0 \][/tex]
Factoring out the common factor:
[tex]\[ x(x + 4) = 0 \][/tex]
Setting each factor to zero gives the solutions:
[tex]\( x = 0 \)[/tex] and [tex]\( x + 4 = 0 \)[/tex] which gives [tex]\( x = -4 \)[/tex]
These solutions indicate the points [tex]\((0, -21)\)[/tex] and [tex]\((-4, -21)\)[/tex] are on the graph.
(d) Graphing [tex]\( f(x) = x^2 + 4x - 21 \)[/tex]:
Using the information:
- Vertex: [tex]\((-2, -25)\)[/tex]
- x-intercepts: [tex]\((-7, 0)\)[/tex] and [tex]\((3, 0)\)[/tex]
- Points from [tex]\( f(x) = -21 \)[/tex]: [tex]\((0, -21)\)[/tex] and [tex]\((-4, -21)\)[/tex]
You can plot these points on a coordinate plane to sketch the graph of the function. The parabola will open upwards, as the coefficient of [tex]\( x^2 \)[/tex] is positive. These critical points help define the curve's shape and location accurately.
