Question

1. for each quadratic function below

i) determine the x-intercepts of the graph
ii) determine the y-intercept of the graph
iii) determine the equation of the axis of symmetry
iv) determine the coordinates of the vertex
v) sketch the graph
a) ( f(x) = (x + 4)(x - 2) )
b) ( g(x) = -2x(x - 3) )
c) ( h(x) = 2(x + 1)(x - 7) )

Explanation:

Part (a): \( f(x) = (x + 4)(x - 2) \)

i) Determine the \( x \)-intercepts

To find the \( x \)-intercepts, set \( f(x) = 0 \):

$$ (x + 4)(x - 2) = 0 $$

This gives \( x + 4 = 0 \) or \( x - 2 = 0 \), so \( x = -4 \) or \( x = 2 \).

ii) Determine the \( y \)-intercept

Set \( x = 0 \) in \( f(x) \):

$$ f(0) = (0 + 4)(0 - 2) = 4 \times (-2) = -8 $$

So the \( y \)-intercept is \( (0, -8) \).

iii) Determine the axis of symmetry

For a quadratic in factored form \( f(x) = (x - r_1)(x - r_2) \), the axis of symmetry is \( x = \frac{r_1 + r_2}{2} \). Here, \( r_1 = -4 \) and \( r_2 = 2 \):

$$ x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1 $$

iv) Determine the vertex coordinates

The axis of symmetry is \( x = -1 \). Substitute \( x = -1 \) into \( f(x) \):

$$ f(-1) = (-1 + 4)(-1 - 2) = 3 \times (-3) = -9 $$

So the vertex is \( (-1, -9) \).

v) Sketch the graph

• Intercepts: \( x \)-intercepts at \( (-4, 0) \) and \( (2, 0) \), \( y \)-intercept at \( (0, -8) \).
• Vertex: \( (-1, -9) \).
• Shape: Since the coefficient of \( x^2 \) (when expanded) is positive, the parabola opens upwards. Plot the intercepts, vertex, and draw a parabola through these points.

Part (b): \( g(x) = -2x(x - 3) \)

i) Determine the \( x \)-intercepts

Set \( g(x) = 0 \):

$$ -2x(x - 3) = 0 $$

This gives \( -2x = 0 \) or \( x - 3 = 0 \), so \( x = 0 \) or \( x = 3 \).

ii) Determine the \( y \)-intercept

Set \( x = 0 \) in \( g(x) \):

$$ g(0) = -2(0)(0 - 3) = 0 $$

So the \( y \)-intercept is \( (0, 0) \) (which is also an \( x \)-intercept).

iii) Determine the axis of symmetry

For \( g(x) = -2x(x - 3) \), the roots are \( r_1 = 0 \) and \( r_2 = 3 \):

$$ x = \frac{0 + 3}{2} = \frac{3}{2} = 1.5 $$

iv) Determine the vertex coordinates

Substitute \( x = 1.5 \) into \( g(x) \):

$$ g(1.5) = -2(1.5)(1.5 - 3) = -3(-1.5) = 4.5 $$

So the vertex is \( (1.5, 4.5) \).

v) Sketch the graph

• Intercepts: \( x \)-intercepts at \( (0, 0) \) and \( (3, 0) \), \( y \)-intercept at \( (0, 0) \).
• Vertex: \( (1.5, 4.5) \).
• Shape: The coefficient of \( x^2 \) (when expanded) is \( -2 \) (negative), so the parabola opens downwards. Plot the intercepts, vertex, and draw a parabola through these points.

Part (c): \( h(x) = 2(x + 1)(x - 7) \)

i) Determine the \( x \)-intercepts

Set \( h(x) = 0 \):

$$ 2(x + 1)(x - 7) = 0 $$

This gives \( x + 1 = 0 \) or \( x - 7 = 0 \), so \( x = -1 \) or \( x = 7 \).

ii) Determine the \( y \)-intercept

Set \( x = 0 \) in \( h(x) \):

$$ h(0) = 2(0 + 1)(0 - 7) = 2 \times 1 \times (-7) = -14 $$

So the \( y \)-intercept is \( (0, -14) \).

iii) Determine the axis of symmetry

For \( h(x) = 2(x + 1)(x - 7) \), the roots are \( r_1 = -1 \) and \( r_2 = 7 \):

$$ x = \frac{-1 + 7}{2} = \frac{6}{2} = 3 $$

iv) Determine the vertex coordinates

Substitute \( x = 3 \) into \( h(x) \):

$$ h(3) = 2(3 + 1)(3 - 7) = 2 \times 4 \times (-4) = -32 $$

So the vertex is \( (3, -32) \).

v) Sketch the graph

• Intercepts: \( x \)-intercepts at \( (-1, 0) \) and \( (7, 0) \), \( y \)-intercept at \( (0, -14) \).
• Vertex: \( (3, -32) \).
• Shape: The coefficient of \( x^2 \) (when expanded) is \( 2 \) (positive), so the parabola opens upwards. Plot the intercepts, vertex, and draw a parabola through these points.

Answer:

Part (a): \( f(x) = (x + 4)(x - 2) \)

i) Determine the \( x \)-intercepts

To find the \( x \)-intercepts, set \( f(x) = 0 \):

$$ (x + 4)(x - 2) = 0 $$

This gives \( x + 4 = 0 \) or \( x - 2 = 0 \), so \( x = -4 \) or \( x = 2 \).

ii) Determine the \( y \)-intercept

Set \( x = 0 \) in \( f(x) \):

$$ f(0) = (0 + 4)(0 - 2) = 4 \times (-2) = -8 $$

So the \( y \)-intercept is \( (0, -8) \).

iii) Determine the axis of symmetry

For a quadratic in factored form \( f(x) = (x - r_1)(x - r_2) \), the axis of symmetry is \( x = \frac{r_1 + r_2}{2} \). Here, \( r_1 = -4 \) and \( r_2 = 2 \):

$$ x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1 $$

iv) Determine the vertex coordinates

The axis of symmetry is \( x = -1 \). Substitute \( x = -1 \) into \( f(x) \):

$$ f(-1) = (-1 + 4)(-1 - 2) = 3 \times (-3) = -9 $$

So the vertex is \( (-1, -9) \).

v) Sketch the graph

• Intercepts: \( x \)-intercepts at \( (-4, 0) \) and \( (2, 0) \), \( y \)-intercept at \( (0, -8) \).
• Vertex: \( (-1, -9) \).
• Shape: Since the coefficient of \( x^2 \) (when expanded) is positive, the parabola opens upwards. Plot the intercepts, vertex, and draw a parabola through these points.

Part (b): \( g(x) = -2x(x - 3) \)

i) Determine the \( x \)-intercepts

Set \( g(x) = 0 \):

$$ -2x(x - 3) = 0 $$

This gives \( -2x = 0 \) or \( x - 3 = 0 \), so \( x = 0 \) or \( x = 3 \).

ii) Determine the \( y \)-intercept

Set \( x = 0 \) in \( g(x) \):

$$ g(0) = -2(0)(0 - 3) = 0 $$

So the \( y \)-intercept is \( (0, 0) \) (which is also an \( x \)-intercept).

iii) Determine the axis of symmetry

For \( g(x) = -2x(x - 3) \), the roots are \( r_1 = 0 \) and \( r_2 = 3 \):

$$ x = \frac{0 + 3}{2} = \frac{3}{2} = 1.5 $$

iv) Determine the vertex coordinates

Substitute \( x = 1.5 \) into \( g(x) \):

$$ g(1.5) = -2(1.5)(1.5 - 3) = -3(-1.5) = 4.5 $$

So the vertex is \( (1.5, 4.5) \).

v) Sketch the graph

• Intercepts: \( x \)-intercepts at \( (0, 0) \) and \( (3, 0) \), \( y \)-intercept at \( (0, 0) \).
• Vertex: \( (1.5, 4.5) \).
• Shape: The coefficient of \( x^2 \) (when expanded) is \( -2 \) (negative), so the parabola opens downwards. Plot the intercepts, vertex, and draw a parabola through these points.

Part (c): \( h(x) = 2(x + 1)(x - 7) \)

i) Determine the \( x \)-intercepts

Set \( h(x) = 0 \):

$$ 2(x + 1)(x - 7) = 0 $$

This gives \( x + 1 = 0 \) or \( x - 7 = 0 \), so \( x = -1 \) or \( x = 7 \).

ii) Determine the \( y \)-intercept

Set \( x = 0 \) in \( h(x) \):

$$ h(0) = 2(0 + 1)(0 - 7) = 2 \times 1 \times (-7) = -14 $$

So the \( y \)-intercept is \( (0, -14) \).

iii) Determine the axis of symmetry

For \( h(x) = 2(x + 1)(x - 7) \), the roots are \( r_1 = -1 \) and \( r_2 = 7 \):

$$ x = \frac{-1 + 7}{2} = \frac{6}{2} = 3 $$

iv) Determine the vertex coordinates

Substitute \( x = 3 \) into \( h(x) \):

$$ h(3) = 2(3 + 1)(3 - 7) = 2 \times 4 \times (-4) = -32 $$

So the vertex is \( (3, -32) \).

v) Sketch the graph

• Intercepts: \( x \)-intercepts at \( (-1, 0) \) and \( (7, 0) \), \( y \)-intercept at \( (0, -14) \).
• Vertex: \( (3, -32) \).
• Shape: The coefficient of \( x^2 \) (when expanded) is \( 2 \) (positive), so the parabola opens upwards. Plot the intercepts, vertex, and draw a parabola through these points.