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QUESTION 3

Given: [tex]\( g(x) = x^2 - 5x - 6 \)[/tex]

1. Where does the graph of this function intersect the y-axis?
2. Determine the coordinates of the x-intercepts of the graph.
3. Determine the equation of the axis of symmetry of the graph.
4. Hence, give the coordinates of the turning point of the graph.
5. Write down the range of [tex]\( g \)[/tex].
6. On the same set of axes, draw the graph of the function [tex]\( f \)[/tex] defined by [tex]\( y = 2x - 6 \)[/tex] and [tex]\( g \)[/tex].
7. Show on the graph, using points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], where [tex]\( f(x) = g(x) \)[/tex] and calculate the coordinates of these points. Show ALL relevant workings.
8. Hence, determine the value(s) of [tex]\( x \)[/tex] for which [tex]\( g(x) \leq f(x) \)[/tex].


Sagot :

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]
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