Question

22 analyze the quadratic function below. which of the following statements describe the quadratic function? select all that apply. the function has an extreme value at 1.5. the axis of symmetry is $x = \frac{3}{2}$. the vertex of the parabola is a minimum point. the reflection of the point marked on the parabola over the axis of symmetry is $(-2,1)$

Explanation:

Step 1: Analyze the direction of the parabola

The parabola opens downward (since it curves downward), so the vertex is a maximum point. Thus, the statement "The vertex of the parabola is a minimum point" is incorrect.

Step 2: Determine the axis of symmetry

The axis of symmetry of a parabola is the vertical line through its vertex. For a parabola, the axis of symmetry is given by \( x = h\), where \( h\) is the x - coordinate of the vertex. If the vertex is at \( x=\frac{3}{2}=1.5\), then the axis of symmetry is \( x=\frac{3}{2}\), so this statement is correct.

Step 3: Analyze the "extreme value" statement

The vertex of the parabola is the extreme point (maximum, in this case). The x - coordinate of the vertex is \( \frac{3}{2}=1.5\), so the function has an extreme value (the vertex) at \( x = 1.5\) (the x - location of the extreme point), so this statement is correct.

Step 4: Analyze the reflection statement

The reflection of a point \((x,y)\) over the vertical line \( x = h\) is given by \((2h - x,y)\). If the axis of symmetry is \( x=\frac{3}{2}\) and the marked point is, say, \((2,y)\), then the reflected point is \(2\times\frac{3}{2}-2=3 - 2 = 1\), so the reflected point should be \((1,y)\), not \((-2,1)\). Thus, this statement is incorrect.

So the correct statements are: "The function has an extreme value at 1.5" and "The axis of symmetry is \( x=\frac{3}{2} \)".

Answer:

The axis of symmetry is \( x = \frac{3}{2} \), The reflection of the point marked on the parabola over the axis of symmetry is \((-2,1)\) (Note: Wait, there might be a typo, probably \((1,1)\) or check the graph again, but assuming the options as given, let's analyze each:

1. The function has an extreme value at 1.5: The vertex's y - value? Wait, no, the axis of symmetry x - value is 1.5 (\(\frac{3}{2}\)). Wait, the extreme value (vertex) has x - coordinate \(\frac{3}{2}\), but the statement says "extreme value at 1.5" which is ambiguous. But let's check other options.

1. The axis of symmetry is \( x=\frac{3}{2} \): For a parabola, the axis of symmetry is the vertical line through the vertex. If the vertex is at \( x = \frac{3}{2}\) (1.5), this is correct.

1. The vertex of the parabola is a minimum point: The parabola opens downward (since it's a "frown" shape), so the vertex is a maximum point, not minimum. So this is incorrect.

1. The reflection of the point marked on the parabola over the axis of symmetry is \((-2,1)\): Wait, the marked point is (2, - 1)? Wait, the graph has a point (2, - 1)? Wait, the user's graph shows a point (2,1)? Wait, maybe a typo. But assuming the axis of symmetry is \( x=\frac{3}{2}\), the reflection of a point \((x,y)\) over \( x = h\) is \((2h - x,y)\). If the point is (2, - 1) and \( h=\frac{3}{2}\), then the reflection is \(2\times\frac{3}{2}-2=3 - 2 = 1\), so the reflected point is (1, - 1). But the option says (-2,1), which is incorrect. Wait, maybe the marked point is (2,1). Then reflection over \( x=\frac{3}{2}\) is \(2\times\frac{3}{2}-2 = 3 - 2=1\), so (1,1). But the option says (-2,1), which is wrong. But maybe the original problem has a different point. However, based on the options, the correct ones are:

• The axis of symmetry is \( x=\frac{3}{2} \) (assuming the vertex is at \( x=\frac{3}{2}\))
• (If the extreme value statement is about the x - coordinate of the vertex, then "The function has an extreme value at 1.5" could be correct as the vertex (extreme point) has x - coordinate 1.5)

Wait, let's re - evaluate:

1. The function has an extreme value at 1.5: The extreme value (vertex) occurs at \( x = 1.5\) (the x - coordinate of the vertex), so this is correct (the extreme value's x - location is 1.5).

1. The axis of symmetry is \( x=\frac{3}{2} \): Correct, as axis of symmetry is \( x=\frac{3}{2}\).

1. The vertex of the parabola is a minimum point: Incorrect, opens downward, vertex is maximum.

1. The reflection of the point marked on the parabola over the axis of symmetry is \((-2,1)\): Incorrect, as calculated earlier.

So the correct options are "The function has an extreme value at 1.5" and "The axis of symmetry is \( x=\frac{3}{2} \)".