Question

1 classify each graph or equation as a linear, quadratic, or exponential function.
function representation linear quadratic exponential
$y = \frac{1}{2}x - 5$
$y = 7^x + 2$
$y = 2x^2 - 5x$
graphs: a parabola opening upwards, a straight line with negative slope, a curve decreasing towards the x - axis as x increases

Explanation:

For \( y = \frac{1}{2}x - 5 \):

Step1: Recall linear function form

A linear function has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.
The equation \( y=\frac{1}{2}x - 5 \) is in the form \( y=mx + b \) with \( m=\frac{1}{2} \) and \( b=- 5 \). So it is a linear function.

For \( y = 7^{x}+2 \):

Step1: Recall exponential function form

An exponential function has the form \( y = a^{x}+k \) (where \( a>0,a
eq1 \)).
The equation \( y = 7^{x}+2 \) is in the form of an exponential function with \( a = 7 \) and \( k = 2 \). So it is an exponential function.

For \( y=2x^{2}-5x \):

Step1: Recall quadratic function form

A quadratic function has the form \( y=ax^{2}+bx + c\) (where \( a
eq0 \)).
The equation \( y = 2x^{2}-5x \) is in the form \( y=ax^{2}+bx + c\) with \( a = 2\), \( b=-5 \) and \( c = 0 \). So it is a quadratic function.

For the parabola - shaped graph:

Step1: Recall quadratic graph shape

The graph of a quadratic function is a parabola (a U - shaped or inverted U - shaped curve). The given graph that looks like a parabola is the graph of a quadratic function.

For the straight - line graph:

Step1: Recall linear graph shape

The graph of a linear function is a straight line. The given graph that is a straight line is the graph of a linear function.

For the graph with exponential growth/decay shape:

Step1: Recall exponential graph shape

The graph of an exponential function has a characteristic curve that either grows rapidly (for \( a>1 \)) or decays rapidly (for \( 0 < a<1 \)) as \( x \) changes. The given graph with the curve that shows exponential behavior is the graph of an exponential function.

Classification Table:

Function Representation	Linear	Quadratic	Exponential
\( y = 7^{x}+2 \)			$\boldsymbol{\circ}$
\( y=2x^{2}-5x \)		$\boldsymbol{\circ}$
Parabola - shaped graph		$\boldsymbol{\circ}$
Straight - line graph	$\boldsymbol{\circ}$
Exponential - curve graph			$\boldsymbol{\circ}$

Answer:

For \( y = \frac{1}{2}x - 5 \):

Step1: Recall linear function form

A linear function has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.
The equation \( y=\frac{1}{2}x - 5 \) is in the form \( y=mx + b \) with \( m=\frac{1}{2} \) and \( b=- 5 \). So it is a linear function.

For \( y = 7^{x}+2 \):

Step1: Recall exponential function form

An exponential function has the form \( y = a^{x}+k \) (where \( a>0,a
eq1 \)).
The equation \( y = 7^{x}+2 \) is in the form of an exponential function with \( a = 7 \) and \( k = 2 \). So it is an exponential function.

For \( y=2x^{2}-5x \):

Step1: Recall quadratic function form

A quadratic function has the form \( y=ax^{2}+bx + c\) (where \( a
eq0 \)).
The equation \( y = 2x^{2}-5x \) is in the form \( y=ax^{2}+bx + c\) with \( a = 2\), \( b=-5 \) and \( c = 0 \). So it is a quadratic function.

For the parabola - shaped graph:

Step1: Recall quadratic graph shape

The graph of a quadratic function is a parabola (a U - shaped or inverted U - shaped curve). The given graph that looks like a parabola is the graph of a quadratic function.

For the straight - line graph:

Step1: Recall linear graph shape

The graph of a linear function is a straight line. The given graph that is a straight line is the graph of a linear function.

For the graph with exponential growth/decay shape:

Step1: Recall exponential graph shape

The graph of an exponential function has a characteristic curve that either grows rapidly (for \( a>1 \)) or decays rapidly (for \( 0 < a<1 \)) as \( x \) changes. The given graph with the curve that shows exponential behavior is the graph of an exponential function.

Classification Table:

Function Representation	Linear	Quadratic	Exponential
\( y = 7^{x}+2 \)			$\boldsymbol{\circ}$
\( y=2x^{2}-5x \)		$\boldsymbol{\circ}$
Parabola - shaped graph		$\boldsymbol{\circ}$
Straight - line graph	$\boldsymbol{\circ}$
Exponential - curve graph			$\boldsymbol{\circ}$