Graph Intercepts & Symmetry

Graph Intercepts

Intercepts are points where a graph crosses or touches an axis.

• x-intercept(s): Point(s) where the graph crosses the x-axis ($y = 0$). Represented as $(x, 0)$.

• y-intercept(s): Point(s) where the graph crosses the y-axis ($x = 0$). Represented as $(0, y)$.

Interpreting Intercepts and Maxima in Context

When analyzing graphs representing real-world scenarios:

• The y-intercept $(0, y)$ typically represents the starting value or initial condition (e.g., initial height when horizontal distance is zero).

• The x-intercept $(x, 0)$ typically represents a final state or a specific condition when the dependent variable is zero (e.g., total horizontal distance traveled when height is zero).

• A maximum point $(x<em>{max}, y</em>{max})$ indicates the peak value of the dependent variable and the corresponding independent variable at which it occurs (e.g., maximum height and the horizontal distance at which it's achieved).

Finding Intercepts Algebraically

To find intercepts for an equation:

• x-intercept(s): Set $y=0$ and solve for $x$.

• y-intercept(s): Set $x=0$ and solve for $y$.

Examples:

• For $y = x^2 + 1$:

  • x-intercepts: $0 = x^2 + 1
    ightarrow x^2 = -1$. No real x-intercepts.</p></li><li><p>y-intercept:$y = 0^2 + 1
    ightarrow y = 1$. Intercept:$(0, 1)$.</p></li></ul></li><li><p>For$y = x^2 + 81$:</p><ul><li><p>x-intercepts:$0 = x^2 + 81
    ightarrow x^2 = -81$. No real x-intercepts.</p></li><li><p>y-intercept:$y = 0^2 + 81
    ightarrow y = 81$. Intercept:$(0, 81)$.</p></li></ul></li><li><p>For$y = x^3 - 512$:</p><ul><li><p>x-intercepts:$0 = x^3 - 512
    ightarrow x^3 = 512
    ightarrow x = 8$. Intercept:$(8, 0)$.</p></li><li><p>y-intercept:$y = 0^3 - 512
    ightarrow y = -512$. Intercept:$(0, -512)$.</p></li></ul></li></ul><h4 id="64dfef4d-5ff2-468e-b8f8-0c86caf283dd" data-toc-id="64dfef4d-5ff2-468e-b8f8-0c86caf283dd" collapsed="false" seolevelmigrated="true">Graph Symmetry</h4><p>Symmetry describes how a graph looks the same after certain transformations.</p><ul><li><p><strong>Symmetry with respect to the x-axis</strong>: If$(x, y)$is on the graph, then$(x, -y)$is also on the graph. Test by replacing$y$with$-y$; if the resulting equation is equivalent to the original, it is x-axis symmetric.</p></li><li><p><strong>Symmetry with respect to the y-axis</strong>: If$(x, y)$is on the graph, then$(-x, y)$is also on the graph. Test by replacing$x$with$-x$; if the resulting equation is equivalent to the original, it is y-axis symmetric.</p></li><li><p><strong>Symmetry with respect to the origin</strong>: If$(x, y)$is on the graph, then$(-x, -y)$is also on the graph. Test by replacing$x$with$-x$and$y$with$-y$; if the resulting equation is equivalent to the original, it is origin symmetric.</p></li></ul><p><em>Examples:</em></p><ul><li><p>For$y = x^2 + 81$:</p><ul><li><p>x-axis:$-y = x^2 + 81
    ightarrow y = -x^2 - 81$. No.</p></li><li><p>y-axis:$y = (-x)^2 + 81
    ightarrow y = x^2 + 81$. Yes.</p></li><li><p>Origin:$-y = (-x)^2 + 81
    ightarrow -y = x^2 + 81
    ightarrow y = -x^2 - 81$. No.</p></li></ul></li><li><p>For$y = x^3 - 512$:</p><ul><li><p>x-axis:$-y = x^3 - 512
    ightarrow y = -x^3 + 512$. No.</p></li><li><p>y-axis:$y = (-x)^3 - 512
    ightarrow y = -x^3 - 512$. No.</p></li><li><p>Origin:$-y = (-x)^3 - 512
    ightarrow -y = -x^3 - 512
    ightarrow y = x^3 + 512$$. No.