Question

Graph the function. Label the \(x\)-intercept(s) and the \(y\)-intercept. \(g(x)=(x-2)^2-4\)

Short answer

The \(y\)-intercept is at the point (0,4) and the \(x\)-intercepts are at the points (4,0) and (0,0). The graph is a parabola shifted right 2 units and down 4 units from the origin which opens upwards

Step-by-step solution

Finding the \(y\)-intercept

To find the \(y\)-intercept, set \(x\) to 0 in the equation. Hence, \(g(0) = (0-2)^2-4 = 4\). Thus, the \(y\)-intercept is (0,4)

Finding the \(x\)-intercepts

To find the \(x\)-intercepts, set the whole equation to 0 and solve for \(x\). Hence, \( 0 = (x-2)^2-4 \) which simplifies to \( (x-2)^2 = 4 \). Taking square root on both sides gives \(x-2 = \pm 2 \), and after simplifying, we get \(x = 4,0\) Thus, the \(x\)-intercepts are (4,0) and (0,0)

Plotting the Function

Plot the point (0,4) which is the \(y\)-intercept, and points (4,0) and (0,0) which are the \(x\)-intercepts on the graph. You will realize that the graph is a parabola that opens upwards. Sketch the curve of the parabola accordingly

Key concepts

Graphing Parabolas

Parabolas are the graph representations of quadratic functions, meaning any function that can be written in the form of \( ax^2 + bx + c \). In this exercise, we are dealing specifically with the function \( g(x) = (x-2)^2 - 4 \). This is in vertex form, which is generally \( a(x-h)^2 + k \). Understanding the vertex form allows you to easily identify the vertex of the parabola, which in this case is at the point \((h, k) = (2, -4)\).

Here are basic characteristics of a parabola that are helpful when graphing:

• The direction it opens: If the coefficient \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
• The vertex: Being the highest or lowest point on the graph depending on the direction the parabola opens.
• The axis of symmetry: A vertical line that runs through the vertex; for the above function, it is \(x = 2\).

By identifying the direction, vertex, and axis of symmetry, you set a solid foundation for graphing the function accurately.

X-Intercepts

The \(x\)-intercepts are points where the graph crosses the x-axis. At these points, the value of the function, \(g(x)\), is zero. To find the x-intercepts for \(g(x) = (x-2)^2 - 4\), we need to solve the equation \((x-2)^2 - 4 = 0\).

Simplifying this equation involves:

• Adding 4 to both sides, resulting in \((x-2)^2 = 4\).
• Taking the square root of both sides, yielding the solutions \(x - 2 = \pm 2\).
• Finally, solving for \(x\): adding 2 results in \(x = 4\) and subtracting 2 gives \(x = 0\).

Thus, the \(x\)-intercepts are at the points \((4, 0)\) and \((0, 0)\). These points help in appropriately plotting the parabola on the coordinate plane, providing guidance on how the curve bends and where it influences the x-axis.

Y-Intercepts

The \(y\)-intercept of a graph is the point where the graph crosses the y-axis. At this specific point, the value of \(x\) is zero. To determine the \(y\)-intercept for the quadratic function \(g(x) = (x-2)^2 - 4\), we substitute \(x = 0\) into the equation.

The calculation goes as follows:

• Substitute 0 for \(x\): \(g(0) = (0-2)^2 - 4\).
• Simplify the expression to \((0-2)^2 - 4 = 4 - 4\).
• Conclude with the result: \(g(0) = 0\), hence the \(y\)-intercept is at the point \((0, 0)\).

This result shows a particular situation where the \(y\)-intercept is also one of the \(x\)-intercepts. Plotting this point on the graph provides essential information for shaping the parabola correctly.