QUESTION 3

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

3.1 Where does the graph of this function intersect the y-axis?

3.2 Determine the coordinates of the x-intercepts of the graph.

3.3 Determine the equation of the axis of symmetry of the graph. Hence, give the coordinates of the turning point of the graph.

3.4 Write down the range of [tex]\( g \)[/tex].

3.5 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].

3.6 Show on the graph, using A and B, where [tex]\( f(x) = g(x) \)[/tex] and calculate the coordinates of these points. Show ALL relevant workings.

3.7 Hence, determine the value(s) of [tex]\( x \)[/tex] for which [tex]\( g(x) \leq f(x) \)[/tex].



Answer :

Sure! Let's solve each part of the question one by one.

### 3.1 Intersection with the y-axis

To find where the graph intersects the y-axis, we set [tex]\(x = 0\)[/tex]:

[tex]\[ g(0) = 0^2 - 5 \cdot 0 - 6 = -6 \][/tex]

Answer: The graph intersects the y-axis at [tex]\((0, -6)\)[/tex].

### 3.2 Coordinates of the x-intercepts

To find the x-intercepts, we set [tex]\(g(x) = 0\)[/tex]:

[tex]\[ x^2 - 5x - 6 = 0 \][/tex]

We solve this quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -6\)[/tex]:

[tex]\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{25 + 24}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{49}}{2} \][/tex]
[tex]\[ x = \frac{5 \pm 7}{2} \][/tex]

Thus, the solutions are:

[tex]\[ x = \frac{12}{2} = 6 \][/tex]
[tex]\[ x = \frac{-2}{2} = -1 \][/tex]

Answer: The x-intercepts are at [tex]\((6, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].

### 3.3 Equation of the axis of symmetry and turning point

The axis of symmetry for a quadratic function [tex]\(ax^2 + bx + c\)[/tex] is given by:

[tex]\[ x = -\frac{b}{2a} \][/tex]

For the given function [tex]\(g(x) = x^2 - 5x - 6\)[/tex]:

[tex]\[ x = -\frac{-5}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
[tex]\[ x = 2.5 \][/tex]

To find the turning point, we substitute [tex]\( x = 2.5 \)[/tex] back into the function:

[tex]\[ g(2.5) = (2.5)^2 - 5 \cdot 2.5 - 6 \][/tex]
[tex]\[ g(2.5) = 6.25 - 12.5 - 6 \][/tex]
[tex]\[ g(2.5) = -12.25 \][/tex]

Answer: The axis of symmetry is [tex]\(x = 2.5\)[/tex]. The turning point is [tex]\((2.5, -12.25)\)[/tex].

### 3.4 Range of [tex]\(g\)[/tex]

Since [tex]\(g(x) = x^2 - 5x - 6\)[/tex] is a quadratic function opening upwards (because the coefficient of [tex]\(x^2\)[/tex] is positive), the range is all values [tex]\(y \geq \)[/tex] the minimum value found at the turning point.

Answer: The range of [tex]\(g\)[/tex] is [tex]\([-12.25, \infty)\)[/tex].

### 3.5 Graph of [tex]\(f(x) = 2x - 6\)[/tex]

The equation given is [tex]\(y - 2x = -6\)[/tex]. We can rearrange it to the slope-intercept form [tex]\(y = mx + c\)[/tex]:

[tex]\[ y = 2x - 6 \][/tex]

### 3.6 Drawing the graph

Let's graph both [tex]\(g(x) = x^2 - 5x - 6\)[/tex] and [tex]\(f(x) = 2x - 6\)[/tex] on the same set of axes. (It's expected to use graphing tools or manually sketch the graphs.)

### 3.7 Intersection Points

To find where [tex]\(f(x) = g(x)\)[/tex]:

[tex]\[ x^2 - 5x - 6 = 2x - 6 \][/tex]
[tex]\[ x^2 - 5x - 6 - 2x + 6 = 0 \][/tex]
[tex]\[ x^2 - 7x = 0 \][/tex]
[tex]\[ x(x - 7) = 0 \][/tex]

Thus, the solutions are:

[tex]\[ x = 0 \text{ or } x = 7 \][/tex]

Substitute these [tex]\(x\)[/tex] values into [tex]\(f(x)\)[/tex] or [tex]\(g(x)\)[/tex] to find [tex]\(y\)[/tex]:

For [tex]\(x = 0\)[/tex]:

[tex]\[ y = 2(0) - 6 = -6 \][/tex]
So, one intersection point is [tex]\((0, -6)\)[/tex].

For [tex]\(x = 7\)[/tex]:

[tex]\[ y = 2(7) - 6 = 14 - 6 = 8 \][/tex]
So, the other intersection point is [tex]\((7, 8)\)[/tex].

Answer: The points are [tex]\(A(0, -6)\)[/tex] and [tex]\(B(7, 8)\)[/tex].

### 3.8 Solving [tex]\(g(x) \leq f(x)\)[/tex]

We need to find the values of [tex]\(x\)[/tex] where:

[tex]\[ x^2 - 7x \leq 0 \][/tex]

This inequality can be solved by finding the sign changes around the roots of [tex]\(x(x - 7) = 0\)[/tex]:

The roots are [tex]\(x = 0\)[/tex] and [tex]\(x = 7\)[/tex]. Test intervals around these points:
1. [tex]\( x < 0 \)[/tex]: Choose [tex]\(x = -1\)[/tex]. [tex]\( (-1)(-8) = 8 > 0\)[/tex]
2. [tex]\( 0 < x < 7 \)[/tex]: Choose [tex]\(x = 1\)[/tex]. [tex]\( (1)(-6) = -6 < 0\)[/tex]
3. [tex]\( x > 7 \)[/tex]: Choose [tex]\(x = 8\)[/tex]. [tex]\( (8)(1) = 8 > 0\)[/tex]

Thus, [tex]\(g(x) \leq f(x)\)[/tex] in the interval:

Answer: The values of [tex]\(x\)[/tex] are [tex]\( x \in [0, 7] \)[/tex].