Sure, let’s go through each part of the question step-by-step.
### Part 3.1: Intersection with the y-axis
The graph of the function [tex]\( g(x) = x^2 - 5x - 6 \)[/tex] intersects the y-axis where [tex]\( x = 0 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ g(0) = 0^2 - 5 \cdot 0 - 6 = -6 \][/tex]
So, the graph intersects the y-axis at the point [tex]\( (0, -6) \)[/tex].
### Part 3.2: Coordinates of the x-intercepts
The x-intercepts occur where [tex]\( g(x) = 0 \)[/tex]. Solve the equation:
[tex]\[ x^2 - 5x - 6 = 0 \][/tex]
Factorizing, we get:
[tex]\[ (x + 1)(x - 6) = 0 \][/tex]
Thus, the solutions are:
[tex]\[ x = -1 \quad \text{or} \quad x = 6 \][/tex]
So, the coordinates of the x-intercepts are [tex]\( (-1, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex].
### Part 3.3: Equation of the axis of symmetry
For a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], the axis of symmetry is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In our function [tex]\( g(x) = x^2 - 5x - 6 \)[/tex], [tex]\( a = 1 \)[/tex] and [tex]\( b = -5 \)[/tex]. Therefore:
[tex]\[ x = -\frac{-5}{2 \cdot 1} = \frac{5}{2} = 2.5 \][/tex]
So, the equation of the axis of symmetry is [tex]\( x = 2.5 \)[/tex].
### Part 3.4: Coordinates of the turning point
The turning point (vertex) is at [tex]\( x = 2.5 \)[/tex]. We substitute [tex]\( x = 2.5 \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[ g(2.5) = (2.5)^2 - 5 \cdot 2.5 - 6 = 6.25 - 12.5 - 6 = -12.25 \][/tex]
Thus, the coordinates of the turning point are [tex]\( (2.5, -12.25) \)[/tex].
### Part 3.5: Range of [tex]\( g \)[/tex]
Since the parabola opens upwards and the lowest point on the graph is the turning point, the range of [tex]\( g(x) \)[/tex] is from the y-value of the turning point to infinity:
[tex]\[ \text{Range of } g(x) = [-12.25, \infty) \][/tex]
### Part 3.6: Drawing the graph
To draw the graph of the function [tex]\( f(x) \)[/tex]:
[tex]\[ y = 2x - 6 \][/tex]
Rewrite it in the form [tex]\( y = 2x - 6 \)[/tex].
### Part 3.7: Intersection points of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
To find where [tex]\( f(x) = g(x) \)[/tex], we solve the equation:
[tex]\[ 2x - 6 = x^2 - 5x - 6 \][/tex]
Rearranging gives us:
[tex]\[ x^2 - 7x = 0 \][/tex]
Factoring, we get:
[tex]\[ x(x - 7) = 0 \][/tex]
So, [tex]\( x = 0 \)[/tex] and [tex]\( x = 7 \)[/tex].
Substitute these back into [tex]\( f(x) \)[/tex] to get the y-coordinates:
When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2 \cdot 0 - 6 = -6 \][/tex]
When [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 2 \cdot 7 - 6 = 14 - 6 = 8 \][/tex]
Thus, the coordinates of the intersection points are [tex]\( A(0, -6) \)[/tex] and [tex]\( B(7, 8) \)[/tex].
### Part 3.8: Values of [tex]\( x \)[/tex] for which [tex]\( g(x) \leq f(x) \)[/tex]
To determine the range of [tex]\( x \)[/tex] where [tex]\( g(x) \leq f(x) \)[/tex]:
We need to solve:
[tex]\[ x^2 - 5x - 6 \leq 2x - 6 \][/tex]
Rearranging:
[tex]\[ x^2 - 7x \leq 0 \][/tex]
Factoring:
[tex]\[ x(x - 7) \leq 0 \][/tex]
This inequality holds where [tex]\( 0 \leq x \leq 7 \)[/tex].
Summarizing all the values:
1. y-intercept: [tex]\( (0, -6) \)[/tex]
2. x-intercepts: [tex]\( (-1, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex]
3. Axis of symmetry: [tex]\( x = 2.5 \)[/tex]
4. Turning point: [tex]\( (2.5, -12.25) \)[/tex]
5. Range of [tex]\( g \)[/tex]: [tex]\([-12.25, \infty)\)[/tex]
6. Intersection points: [tex]\( A(0, -6) \)[/tex] and [tex]\( B(7, 8) \)[/tex]
7. Values of [tex]\( x \)[/tex] for which [tex]\( g(x) \leq f(x) \)[/tex]: [tex]\([0, 7]\)[/tex]
