Free Math Worksheets

- Locate the vertex of each V-shaped graph. - Horizontal translations change the expression inside the absolute value. - Vertical translations add or subtract a constant outside the absolute value.

1. The vertex of \(f\) is \((0, 0)\), and the vertex of \(g\) is \((-3, 1)\). 2. Moving the vertex from \((0, 0)\) to \((-3, 1)\) requires a shift left \(3\) units and up \(1\) unit. 3. A shift left \(3\) units replaces \(x\) with \(x+3\), and a shift up \(1\) unit adds \(1\) outside the absolute value. 4. Therefore, \(g(x)=|x+3|+1\).

Shift left \(3\) units and up \(1\) unit; \(g(x)=|x+3|+1\)

- Rewrite the absolute value function as two linear pieces if helpful. - An x-intercept occurs where \(g(x)=0\). - Use the fact that \(|x|\ge0\).

1. The graph of \(g(x)=4-|x|\) is an upside-down V with vertex \((0, 4)\). For \(x\ge0\), \(g(x)=4-x\); for \(x<0\), \(g(x)=4+x\). 2. Set \(4-|x|=0\). Then \(|x|=4\), so \(x=4\) or \(x=-4\). The x-intercepts are \((4, 0)\) and \((-4, 0)\). 3. Because \(|x|\ge0\), \(4-|x|\le4\). Equality occurs at \(x=0\), so the maximum point is \((0, 4)\).

a) An upside-down V with vertex \((0, 4)\) b) \((-4, 0)\) and \((4, 0)\) c) The maximum point is \((0, 4)\).

- Use a point on the red graph to determine its coefficient. - Write a vertical scaling in the form \(af(x)\). - Write a horizontal scaling in the form \(f(bx)\). - Use the rule \(|ab|=|a||b|\) to compare the two expressions.

1. The red graph passes through \((2, 1)\). Using \(g(x)=a|x|\), substitute the point: \(1=a|2|\). Therefore, \(a=\frac{1}{2}\), so \(g(x)=\frac{1}{2}|x|\). 2. Tim describes a vertical compression by a factor of \(\frac{1}{2}\): \(g(x)=\frac{1}{2}f(x)\). 3. Sara describes a horizontal stretch by a factor of \(2\): \(g(x)=f\left(\frac{x}{2}\right)=\left|\frac{x}{2}\right|=\frac{1}{2}|x|\). 4. Both transformations produce the same function because absolute value is homogeneous for a positive scale factor.

1) \(g(x)=\frac{1}{2}|x|\) 2) Vertical compression factor: \(\frac{1}{2}\), since \(g(x)=\frac{1}{2}f(x)\); horizontal stretch factor: \(2\), since \(g(x)=f\left(\frac{x}{2}\right)\)

- Compare the vertices of the two V-shaped graphs. - Determine whether the red graph opens upward or downward. - Apply the horizontal change inside the input and the reflection outside.

1. The vertex of \(f\) is \((-4, 0)\), and the vertex of \(g\) is \((-2, 0)\). Therefore, the graph shifts right \(2\) units. 2. The blue graph opens upward, while the red graph opens downward, so the shifted graph is reflected across the x-axis. 3. A shift right \(2\) units gives \(f(x-2)\), and the reflection places a negative sign outside. 4. Therefore, \(g(x)=-f(x-2)=-|x+2|\).

Shift right \(2\) units and reflect across the x-axis; \(g(x)=-f(x-2)\)