the graph of $y = f(x)$ is shown below. find all values of $x$ where $f(x) = -7$.

the graph of $y = f(x)$ is shown below. find all values of $x$ where $f(x) = -7$.

the graph of $y = f(x)$ is shown below. find all values of $x$ where $f(x) = -7$.

Answer

Explanation:

Step1: Understand the problem

We need to find the ( x )-values where ( f(x) = -7 ). This means we look for the points on the graph of ( y = f(x) ) where the ( y )-coordinate is ( -7 ), and then find the corresponding ( x )-coordinate(s).

Step2: Analyze the graph

The graph is a straight line (linear function). To find where ( y = -7 ), we can either:

  1. Find the equation of the line and solve for ( x ) when ( y = -7 ).
  2. Visually locate ( y = -7 ) on the ( y )-axis and find the ( x )-value where the line intersects ( y = -7 ).

Let's use the first method for accuracy.

Step2.1: Find the equation of the line

The line passes through two points:

  • The ( y )-intercept: ( (0, 5) ) (since it crosses the ( y )-axis at ( y = 5 ) when ( x = 0 )).
  • The ( x )-intercept: ( (4, 0) ) (since it crosses the ( x )-axis at ( x = 4 ) when ( y = 0 )).

The slope ( m ) of the line is given by: [ m = \frac{y_2 - y_1}{x_2 - x_1} ] Using ( (x_1, y_1) = (0, 5) ) and ( (x_2, y_2) = (4, 0) ): [ m = \frac{0 - 5}{4 - 0} = \frac{-5}{4} = -\frac{5}{4} ]

The equation of the line in slope-intercept form ( y = mx + b ) (where ( b ) is the ( y )-intercept) is: [ y = -\frac{5}{4}x + 5 ]

Step2.2: Solve for ( x ) when ( y = -7 )

Substitute ( y = -7 ) into the equation: [ -7 = -\frac{5}{4}x + 5 ]

Subtract 5 from both sides: [ -7 - 5 = -\frac{5}{4}x ] [ -12 = -\frac{5}{4}x ]

Multiply both sides by ( -\frac{4}{5} ) to solve for ( x ): [ x = (-12) \times \left(-\frac{4}{5}\right) = \frac{48}{5} = 9.6 ]

Wait, that doesn't seem right. Wait, maybe I made a mistake in the slope? Wait, let's check the graph again. Wait, the line goes from the top left to the bottom right, so slope is negative. Wait, when ( x = 0 ), ( y = 5 ); when ( x = 4 ), ( y = 0 ). Then when ( x ) increases, ( y ) decreases. So the slope is ( (0 - 5)/(4 - 0) = -5/4 ), which is correct. Then when ( y = -7 ), solving:

( -7 = (-5/4)x + 5 )

Subtract 5: ( -12 = (-5/4)x )

Multiply both sides by ( -4/5 ): ( x = (-12) \times (-4/5) = 48/5 = 9.6 ). But looking at the graph, the grid lines: each square is 1 unit. Let's check the ( y )-axis: ( y = -7 ) is 7 units below the origin. Let's see the ( x )-axis: from ( x = 0 ) to ( x = 10 ), the line goes from ( (0,5) ) to ( (4,0) ), then continues. Wait, maybe the graph is drawn with integer coordinates? Wait, maybe I misread the graph. Wait, the graph has a grid where each square is 1 unit. Let's check the line: when ( x = 0 ), ( y = 5 ); when ( x = 4 ), ( y = 0 ); when ( x = 8 ), ( y = -5 ); when ( x = 12 ), ( y = -10 ), but the graph only goes to ( x = 10 ). Wait, maybe my initial assumption of the points is wrong. Wait, maybe the line passes through ( (0, 5) ) and ( (4, 0) ), but let's check the slope again. Wait, the line from ( (0,5) ) to ( (4,0) ): the run is 4, rise is -5, so slope is -5/4. Then when ( y = -7 ), let's solve:

( y = -5/4 x + 5 )

( -7 = -5/4 x + 5 )

( -12 = -5/4 x )

( x = (-12) * (-4/5) = 48/5 = 9.6 ). But 9.6 is 48/5, which is 9 and 3/5. But looking at the graph, the ( x )-axis goes up to 10, so ( x = 9.6 ) is between 9 and 10. But maybe the graph is drawn with a different scale? Wait, maybe I made a mistake in the ( y )-intercept. Wait, looking at the graph: the ( y )-axis at ( x = 0 ), the line crosses at ( y = 5 ), correct. Then when ( x = 4 ), ( y = 0 ), correct. Then when ( x = 8 ), ( y = -5 ), because from ( x = 4 ) to ( x = 8 ), that's 4 units, so ( y ) decreases by 5 (from 0 to -5). Then from ( x = 8 ) to ( x = 12 ), ( y ) decreases by 5 to -10. So ( y = -7 ) is 2 units below ( y = -5 ) (since -5 to -7 is -2). So from ( x = 8 ), we need to move ( x ) by ( (2 / 5) * 4 = 8/5 = 1.6 ) units (since the slope is -5/4, so for a change in ( y ) of -2, the change in ( x ) is ( (-2) / (-5/4) = 8/5 = 1.6 )). So ( x = 8 + 1.6 = 9.6 ), which matches the earlier calculation. So the ( x )-value where ( f(x) = -7 ) is ( x = 48/5 = 9.6 ) or ( x = 9.6 ).

Alternatively, by visual inspection: find ( y = -7 ) on the ( y )-axis (7 units below the origin). Then move horizontally until we hit the line. The line at ( y = -7 ) should be at ( x = 9.6 ) (or 48/5).

Wait, but maybe the graph is drawn with a different slope? Wait, let's check the line again. From ( (0,5) ) to ( (4,0) ): that's a slope of -5/4. So the equation is correct. So solving ( y = -7 ) gives ( x = 48/5 = 9.6 ).

But let's confirm with the graph. The grid: each square is 1 unit. So ( y = -7 ) is at the 7th line below the ( x )-axis. The line passes through ( (0,5) ), ( (4,0) ), ( (8,-5) ), ( (12,-10) ). So between ( x = 8 ) (where ( y = -5 )) and ( x = 12 ) (where ( y = -10 )), the line goes from ( y = -5 ) to ( y = -10 ) over ( x = 8 ) to ( x = 12 ) (4 units in ( x ), 5 units in ( y )). So the rate of change is 5 units of ( y ) for 4 units of ( x ). So to go from ( y = -5 ) (at ( x = 8 )) to ( y = -7 ) (which is 2 units down in ( y )), we need to move ( x ) by ( (2 / 5) * 4 = 8/5 = 1.6 ) units. So ( x = 8 + 1.6 = 9.6 ), which is 48/5. So that's correct.

Answer:

The value of ( x ) where ( f(x) = -7 ) is ( \boxed{\dfrac{48}{5}} ) (or ( \boxed{9.6} )).