Questions: Select the graph that shows the relation: x=y^2-5.

Transcript text: Select the graph that shows the relation: $x=y^{2}-5$.

Solution

Solution Steps

To solve the problem of selecting the graph that shows the relation $ x = y^2 - 5 $, we need to understand the shape and orientation of the graph. This equation represents a parabola that opens to the right because $ x $ is expressed in terms of $ y $. The vertex of the parabola is at $( -5, 0 )$.

Step 1: Understand the Equation

The given equation is: \[ x = y^2 - 5 \]

This is a quadratic equation in terms of $ y $. To understand the graph of this equation, we need to analyze its properties.

Step 2: Identify the Shape of the Graph

The equation $ x = y^2 - 5 $ represents a parabola. Since $ y $ is squared, the parabola opens horizontally. Specifically, it opens to the right because the coefficient of $ y^2 $ is positive.

Step 3: Determine the Vertex

The vertex of the parabola can be found by setting $ y = 0 $: \[ x = 0^2 - 5 = -5 \] So, the vertex is at the point $(-5, 0)$.

Step 4: Plot Key Points

To better understand the graph, we can plot a few key points by substituting values for $ y $:

When $ y = 1 $: \[ x = 1^2 - 5 = -4 \] Point: $(-4, 1)$
When $ y = -1 $: \[ x = (-1)^2 - 5 = -4 \] Point: $(-4, -1)$
When $ y = 2 $: \[ x = 2^2 - 5 = -1 \] Point: $(-1, 2)$
When $ y = -2 $: \[ x = (-2)^2 - 5 = -1 \] Point: $(-1, -2)$

Step 5: Sketch the Graph

Using the vertex and the key points, we can sketch the graph. The parabola opens to the right with the vertex at $(-5, 0)$ and passes through the points $(-4, 1)$, $(-4, -1)$, $(-1, 2)$, and $(-1, -2)$.

Final Answer

The graph that shows the relation $ x = y^2 - 5 $ is a horizontally opening parabola with the vertex at $(-5, 0)$. The correct graph will reflect this shape and these key points.

\[ \boxed{\text{The graph is a horizontally opening parabola with vertex at } (-5, 0).} \]

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