Exercise Questions

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This exercise contains 11 questions. Use the Questions tab to view and track them.

Key Concepts

This exercise focuses on the following concepts:

Linear Graphing

A linear function has the form $y = m x + c$ , where:

$m$ = slope (gradient) — measures steepness and direction
$c$ = $y$ -intercept — where the line crosses the $y$ -axis

The slope between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Example: Sketch $y = 2 x - 3$ .

Slope $m = 2$ (rises 2 units for every 1 unit right)
$y$ -intercept: set $x = 0 \Rightarrow y = - 3$ , so point $(0, - 3)$
$x$ -intercept: set $y = 0 \Rightarrow x = 1.5$ , so point $(1.5, 0)$

Quadratic Analysis

A quadratic function has the form $f (x) = a x^{2} + b x + c$ .

If $a > 0$ : parabola opens upward (minimum vertex)
If $a < 0$ : parabola opens downward (maximum vertex)

Vertex and Axis of Symmetry

The vertex is the turning point of the parabola: $x_{vertex} = - \frac{b}{2 a}, y_{vertex} = f (- \frac{b}{2 a})$

The axis of symmetry is the vertical line $x = - \frac{b}{2 a}$ .

Example: Find the vertex of $f (x) = x^{2} - 4 x + 3$ .

$a = 1, b = - 4$
$x = - \frac{- 4}{2 ( 1 )} = 2$
$y = f (2) = 4 - 8 + 3 = - 1$
Vertex: $(2, - 1)$

Intercepts

Intercept	Method
$x$ -intercept(s)	Set $y = 0$ , solve for $x$ (use quadratic formula if needed)
$y$ -intercept	Set $x = 0$ , evaluate $f (0)$

Graphical Solutions

To solve $f (x) = g (x)$ graphically:

Plot both functions on the same axes.
Identify the intersection points.
The $x$ -coordinates of these points are the solutions.

Absolute Value (Modulus) Functions

The function $f (x) = a ∣ x - h ∣ + k$ produces a V-shaped graph:

Vertex at $(h, k)$
If $a > 0$ : V opens upward
If $a < 0$ : V opens downward (inverted V)

Real-World Modeling

Functions are used to model practical situations:

Economic Equilibrium: Supply $S (x)$ and demand $D (x)$ functions intersect at the equilibrium point where supply equals demand.
Projectile Motion: Height $h (t) = - \frac{1}{2} g t^{2} + v_{0} t + h_{0}$ is a downward parabola; the vertex gives maximum height.

Important Formulas

Description	Formula
Slope-Intercept Form	$y = m x + c$
Slope Formula	$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Quadratic Standard Form	$f (x) = a x^{2} + b x + c$
Axis of Symmetry	$x = - \frac{b}{2 a}$
Quadratic Formula	$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Vertex $x$ -coordinate	$x = \frac{x _{1} + x _{2}}{2}$ (midpoint of roots)

Summary

This exercise focuses on the visual representation of algebraic functions on the Cartesian plane.

Linear functions ( $y = m x + c$ ) produce straight lines; identify slope and intercepts to sketch.
Quadratic functions ( $f (x) = a x^{2} + b x + c$ ) produce parabolas; find vertex, axis of symmetry, and intercepts.
Modulus functions ( $f (x) = a ∣ x ∣ + c$ ) produce V-shapes; vertex and direction depend on $a$ .
Graphical solutions are found at intersection points of two curves.
Real-world problems (equilibrium, projectile motion) are solved by identifying key graphical features.

Strategy: Always identify the function type first to predict the graph shape. For quadratics, find the vertex first as the central reference point.

Linear Graphing

A linear function has the form

y = m x + c

, where:

m

= slope (gradient) — measures steepness and direction

c

y

-intercept — where the line crosses the

y

-axis

The slope between two points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

is:

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Example: Sketch

y = 2 x - 3

Slope

m = 2

(rises 2 units for every 1 unit right)

y

-intercept: set

x = 0 \Rightarrow y = - 3

, so point

(0, - 3)

x

-intercept: set

y = 0 \Rightarrow x = 1.5

, so point

(1.5, 0)

Intercept

Method

$x$ -intercept(s)

Set

y = 0

, solve for

x

(use quadratic formula if needed)

$y$ -intercept

Set

x = 0

, evaluate

f (0)

Description

Formula

Slope-Intercept Form

y = m x + c

Slope Formula

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Quadratic Standard Form

f (x) = a x^{2} + b x + c

Axis of Symmetry

x = - \frac{b}{2 a}

Quadratic Formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Vertex $x$ -coordinate

x = \frac{x _{1} + x _{2}}{2}

(midpoint of roots)

Summary

This exercise focuses on the visual representation of algebraic functions on the Cartesian plane.

Linear functions (

y = m x + c

) produce straight lines; identify slope and intercepts to sketch.

Quadratic functions (

f (x) = a x^{2} + b x + c

) produce parabolas; find vertex, axis of symmetry, and intercepts.

Modulus functions (

f (x) = a ∣ x ∣ + c

) produce V-shapes; vertex and direction depend on

a

Graphical solutions are found at intersection points of two curves.

Real-world problems (equilibrium, projectile motion) are solved by identifying key graphical features.

Strategy: Always identify the function type first to predict the graph shape. For quadratics, find the vertex first as the central reference point.

1.2 Exercise 1.2

Exercise Questions

Key Concepts

Linear Graphing

Quadratic Analysis

Vertex and Axis of Symmetry

Intercepts

Graphical Solutions

Absolute Value (Modulus) Functions

Real-World Modeling

Important Formulas

Summary

1.2 Exercise 1.2

Exercise Questions

Key Concepts

Linear Graphing

Quadratic Analysis

Vertex and Axis of Symmetry

Intercepts

Graphical Solutions

Absolute Value (Modulus) Functions

Real-World Modeling

Important Formulas

Summary