Sketch the graph of $y = |2x + a|$, where $a$ is a positive constant - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 3

To sketch the graph of the equation, start by identifying the vertex of the V-shape created by the absolute value function. The expression inside the absolute value, $2x + a$, is equal to zero at the point where:

$2x + a = 0$

Solving this gives:

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

At this point, the vertex of the graph is located at ig(-\frac{a}{2}, 0\big).

Next, since $a$ is a positive constant, we note that the graph will open upwards, creating a V-shape. The graph does not dip below the x-axis since the absolute value function ensures that all output values will be zero or higher. Hence, the two arms of the V will extend infinitely upward from the vertex.

To find where the graph intersects the axes, we can substitute $x=0$ to find the y-intercept:

$y = |2(0) + a| = |a| = a$

Thus, the y-intercept is at $(0, a)$.