Question

drag the tiles to the correct boxes to complete the pairs.
match the graphs with the functions they represent.
graph of parabolas with labels f(x), g(x), h(x), j(x) and functions x² - 3, 2x² - 3, -2x² - 3, x² + 3 to match

Explanation:

Step1: Analyze the vertex and direction of parabolas

For a quadratic function \(y = ax^2 + k\), the vertex is at \((0, k)\) and the direction (upward if \(a>0\), downward if \(a<0\)) and width (narrower if \(|a|>1\), wider if \(|a|<1\)) depend on \(a\).

Step2: Match \(x^2 - 3\)

The function \(y=x^2 - 3\) has \(a = 1>0\) (upward), vertex \((0, - 3)\). Looking at the graphs, \(h(x)\) has vertex \((0,-3)\) and upward opening with \(a = 1\) (since it's the "wider" upward parabola among the red and blue). So \(x^2 - 3\) matches \(h(x)\).

Step3: Match \(2x^2 - 3\)

The function \(y = 2x^2-3\) has \(a=2>0\) (upward), vertex \((0, - 3)\) and narrower than \(y=x^2 - 3\) (since \(|2|>1\)). The red graph \(g(x)\) is narrower than \(h(x)\) and upward, so \(2x^2 - 3\) matches \(g(x)\).

Step4: Match \(-2x^2 - 3\)

The function \(y=-2x^2 - 3\) has \(a=-2<0\) (downward), vertex \((0, - 3)\). The green graph \(j(x)\) is downward opening, so \(-2x^2 - 3\) matches \(j(x)\).

Step5: Match \(x^2 + 3\)

The function \(y=x^2 + 3\) has \(a = 1>0\) (upward), vertex \((0,3)\). The topmost parabola \(f(x)\) has vertex \((0,3)\) and upward opening, so \(x^2 + 3\) matches \(f(x)\).

Answer:

\(x^2 - 3\) → \(h(x)\)
\(2x^2 - 3\) → \(g(x)\)
\(-2x^2 - 3\) → \(j(x)\)
\(x^2 + 3\) → \(f(x)\)